Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.2% → 95.3%

Time: 14.6s

Alternatives: 21

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y - z) * ((t - x) / (a - z)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Python

def code(x, y, z, t, a):
	return x + ((y - z) * ((t - x) / (a - z)))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y - z) * ((t - x) / (a - z)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y - z) * ((t - x) / (a - z)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Python

def code(x, y, z, t, a):
	return x + ((y - z) * ((t - x) / (a - z)))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y - z) * ((t - x) / (a - z)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{t - x}{z} \cdot \left(a - y\right), t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-304)
     t_1
     (if (<= t_2 0.0) (fma 1.0 (* (/ (- t x) z) (- a y)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-304) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(1.0, (((t - x) / z) * (a - y)), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-304)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(1.0, Float64(Float64(Float64(t - x) / z) * Float64(a - y)), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-304], t$95$1, If[LessEqual[t$95$2, 0.0], N[(1.0 * N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999971e-305 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   if -9.99999999999999971e-305 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) + t} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{z} + 1, \frac{t - x}{z} \cdot \left(\left(-y\right) + a\right), t\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{t - x}{z}} \cdot \left(\left(\mathsf{neg}\left(y\right)\right) + a\right), t\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(1, \color{blue}{\frac{t - x}{z}} \cdot \left(\left(-y\right) + a\right), t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{t - x}{z} \cdot \left(a - y\right), t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(a - y, \frac{-x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-304)
     t_1
     (if (<= t_2 0.0) (fma (- a y) (/ (- x) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-304) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((a - y), (-x / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-304)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(a - y), Float64(Float64(-x) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-304], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(a - y), $MachinePrecision] * N[((-x) / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999971e-305 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   if -9.99999999999999971e-305 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 3.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 3.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a - y, \frac{-1 \cdot x}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(a - y, \frac{-x}{z}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - y, \frac{-x}{z}, t\right)\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 0.00068:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a y) (/ (- x) z) t)))
   (if (<= z -4.8e+91)
     t_1
     (if (<= z -7.2e-48)
       (+ t (/ (* y (- x t)) z))
       (if (<= z 0.00068) (fma (/ y a) (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a - y), (-x / z), t);
	double tmp;
	if (z <= -4.8e+91) {
		tmp = t_1;
	} else if (z <= -7.2e-48) {
		tmp = t + ((y * (x - t)) / z);
	} else if (z <= 0.00068) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a - y), Float64(Float64(-x) / z), t)
	tmp = 0.0
	if (z <= -4.8e+91)
		tmp = t_1;
	elseif (z <= -7.2e-48)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	elseif (z <= 0.00068)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - y), $MachinePrecision] * N[((-x) / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4.8e+91], t$95$1, If[LessEqual[z, -7.2e-48], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00068], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.79999999999999966e91 or 6.8e-4 < z

   1. Initial program 67.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a - y, \frac{-1 \cdot x}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 74.5%

       \[\leadsto \mathsf{fma}\left(a - y, \frac{-x}{z}, t\right) \]

   if -4.79999999999999966e91 < z < -7.2000000000000003e-48

   1. Initial program 86.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) + t} \]

   5. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{z} + 1, \frac{t - x}{z} \cdot \left(\left(-y\right) + a\right), t\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right)}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.0%

      \[\leadsto t - \color{blue}{\frac{\left(t - x\right) \cdot y}{z}} \]

   if -7.2000000000000003e-48 < z < 6.8e-4

   1. Initial program 94.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 79.4

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Applied rewrites 79.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(a - y, \frac{-x}{z}, t\right)\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 0.00068:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - y, \frac{-x}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-19}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.3e+15)
   (fma (/ y a) (- t x) x)
   (if (<= a 1.85e-19)
     (+ t (/ (* y (- x t)) z))
     (if (<= a 4e+51) (* (- y z) (/ t (- a z))) (fma (- y z) (/ t a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.3e+15) {
		tmp = fma((y / a), (t - x), x);
	} else if (a <= 1.85e-19) {
		tmp = t + ((y * (x - t)) / z);
	} else if (a <= 4e+51) {
		tmp = (y - z) * (t / (a - z));
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.3e+15)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (a <= 1.85e-19)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	elseif (a <= 4e+51)
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.3e+15], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 1.85e-19], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4e+51], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.3e15

   1. Initial program 93.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 70.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Applied rewrites 70.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   if -2.3e15 < a < 1.85000000000000003e-19

   1. Initial program 74.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) + t} \]

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{z} + 1, \frac{t - x}{z} \cdot \left(\left(-y\right) + a\right), t\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right)}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.7%

      \[\leadsto t - \color{blue}{\frac{\left(t - x\right) \cdot y}{z}} \]

   if 1.85000000000000003e-19 < a < 4e51

   1. Initial program 86.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 86.8

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 65.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]
   8. Step-by-step derivation

   9. Applied rewrites 73.5%

      \[\leadsto \frac{t}{a - z} \cdot \color{blue}{\left(y - z\right)} \]

   if 4e51 < a

   1. Initial program 85.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]

      2. lower--.f64 79.9

         \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t - x}}{a} \]

   5. Applied rewrites 79.9%

      \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 79.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   7. Applied rewrites 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.1%

       \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-19}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+51}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -5.1 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-191}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ z (- z a)))))
   (if (<= z -5.1e-64)
     t_1
     (if (<= z -1.9e-191)
       (/ (* (- y z) t) a)
       (if (<= z 2.05e-7) (* t (/ y (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -5.1e-64) {
		tmp = t_1;
	} else if (z <= -1.9e-191) {
		tmp = ((y - z) * t) / a;
	} else if (z <= 2.05e-7) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z / (z - a))
    if (z <= (-5.1d-64)) then
        tmp = t_1
    else if (z <= (-1.9d-191)) then
        tmp = ((y - z) * t) / a
    else if (z <= 2.05d-7) then
        tmp = t * (y / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -5.1e-64) {
		tmp = t_1;
	} else if (z <= -1.9e-191) {
		tmp = ((y - z) * t) / a;
	} else if (z <= 2.05e-7) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * (z / (z - a))
	tmp = 0
	if z <= -5.1e-64:
		tmp = t_1
	elif z <= -1.9e-191:
		tmp = ((y - z) * t) / a
	elif z <= 2.05e-7:
		tmp = t * (y / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z / Float64(z - a)))
	tmp = 0.0
	if (z <= -5.1e-64)
		tmp = t_1;
	elseif (z <= -1.9e-191)
		tmp = Float64(Float64(Float64(y - z) * t) / a);
	elseif (z <= 2.05e-7)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z / (z - a));
	tmp = 0.0;
	if (z <= -5.1e-64)
		tmp = t_1;
	elseif (z <= -1.9e-191)
		tmp = ((y - z) * t) / a;
	elseif (z <= 2.05e-7)
		tmp = t * (y / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.1e-64], t$95$1, If[LessEqual[z, -1.9e-191], N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 2.05e-7], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.09999999999999984e-64 or 2.05e-7 < z

   1. Initial program 72.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 77.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 41.6

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 49.6%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 49.6%

       \[\leadsto \frac{z}{z - a} \cdot t \]

   if -5.09999999999999984e-64 < z < -1.8999999999999999e-191

   1. Initial program 96.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 45.6

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in a around inf

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 42.3%

       \[\leadsto \frac{\left(y - z\right) \cdot t}{\color{blue}{a}} \]

   if -1.8999999999999999e-191 < z < 2.05e-7

   1. Initial program 94.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 40.8

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{t \cdot y}{\color{blue}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 37.7%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-64}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-191}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{a - y}{z}, t\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- a y) z) t)))
   (if (<= z -5.2e+62)
     t_1
     (if (<= z 0.0007) (fma (/ y (- a z)) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((a - y) / z), t);
	double tmp;
	if (z <= -5.2e+62) {
		tmp = t_1;
	} else if (z <= 0.0007) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(a - y) / z), t)
	tmp = 0.0
	if (z <= -5.2e+62)
		tmp = t_1;
	elseif (z <= 0.0007)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5.2e+62], t$95$1, If[LessEqual[z, 0.0007], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999968e62 or 6.99999999999999993e-4 < z

   1. Initial program 67.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 78.5%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{a - y}{z}}, t\right) \]

   if -5.19999999999999968e62 < z < 6.99999999999999993e-4

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a - z}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a - z}}, t - x, x\right) \]

      2. lower--.f64 84.8

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a - z}}, t - x, x\right) \]

   7. Applied rewrites 84.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a - z}}, t - x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{a - y}{z}, t\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- a y) z) t)))
   (if (<= z -7.2e-48)
     t_1
     (if (<= z 0.0007) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((a - y) / z), t);
	double tmp;
	if (z <= -7.2e-48) {
		tmp = t_1;
	} else if (z <= 0.0007) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(a - y) / z), t)
	tmp = 0.0
	if (z <= -7.2e-48)
		tmp = t_1;
	elseif (z <= 0.0007)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.2e-48], t$95$1, If[LessEqual[z, 0.0007], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2000000000000003e-48 or 6.99999999999999993e-4 < z

   1. Initial program 72.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 77.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.1%

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{a - y}{z}}, t\right) \]

   if -7.2000000000000003e-48 < z < 6.99999999999999993e-4

   1. Initial program 94.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 83.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a y) (/ (- t x) z) t)))
   (if (<= z -7.2e-48)
     t_1
     (if (<= z 0.0007) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a - y), ((t - x) / z), t);
	double tmp;
	if (z <= -7.2e-48) {
		tmp = t_1;
	} else if (z <= 0.0007) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a - y), Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -7.2e-48)
		tmp = t_1;
	elseif (z <= 0.0007)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - y), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.2e-48], t$95$1, If[LessEqual[z, 0.0007], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2000000000000003e-48 or 6.99999999999999993e-4 < z

   1. Initial program 72.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 77.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   if -7.2000000000000003e-48 < z < 6.99999999999999993e-4

   1. Initial program 94.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 83.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - y, \frac{-x}{z}, t\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0007:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- a y) (/ (- x) z) t)))
   (if (<= z -2e+62)
     t_1
     (if (<= z 0.0007) (fma (- y z) (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a - y), (-x / z), t);
	double tmp;
	if (z <= -2e+62) {
		tmp = t_1;
	} else if (z <= 0.0007) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a - y), Float64(Float64(-x) / z), t)
	tmp = 0.0
	if (z <= -2e+62)
		tmp = t_1;
	elseif (z <= 0.0007)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - y), $MachinePrecision] * N[((-x) / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2e+62], t$95$1, If[LessEqual[z, 0.0007], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000007e62 or 6.99999999999999993e-4 < z

   1. Initial program 67.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a - y, \frac{-1 \cdot x}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 73.9%

       \[\leadsto \mathsf{fma}\left(a - y, \frac{-x}{z}, t\right) \]

   if -2.00000000000000007e62 < z < 6.99999999999999993e-4

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(t - x\right) \cdot \left(y - z\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - z\right) \cdot \left(t - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \frac{t - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\frac{t - x}{a}}, x\right) \]

      7. lower--.f64 77.4

         \[\leadsto \mathsf{fma}\left(y - z, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.3e+15)
   (fma (/ y a) (- t x) x)
   (if (<= a 5.4e-34) (+ t (/ (* y (- x t)) z)) (fma (- y z) (/ t a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.3e+15) {
		tmp = fma((y / a), (t - x), x);
	} else if (a <= 5.4e-34) {
		tmp = t + ((y * (x - t)) / z);
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.3e+15)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (a <= 5.4e-34)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.3e+15], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 5.4e-34], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.3e15

   1. Initial program 93.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 70.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Applied rewrites 70.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   if -2.3e15 < a < 5.40000000000000034e-34

   1. Initial program 74.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right)\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} + \frac{a \cdot \left(-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)\right)}{{z}^{2}}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right) + t} \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{z} + 1, \frac{t - x}{z} \cdot \left(\left(-y\right) + a\right), t\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right)}{z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.8%

      \[\leadsto t - \color{blue}{\frac{\left(t - x\right) \cdot y}{z}} \]

   if 5.40000000000000034e-34 < a

   1. Initial program 85.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]

      2. lower--.f64 73.5

         \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t - x}}{a} \]

   5. Applied rewrites 73.5%

      \[\leadsto x + \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a}} + x \]

      4. lower-fma.f64 73.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   7. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)} \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.9%

       \[\leadsto \mathsf{fma}\left(y - z, \frac{t}{\color{blue}{a}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+89)
   (fma a (/ (- t x) z) t)
   (if (<= z 1.05e-5) (fma (/ y a) (- t x) x) (* t (/ z (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+89) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 1.05e-5) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t * (z / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+89)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 1.05e-5)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(t * Float64(z / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+89], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.05e-5], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.99999999999999996e89

   1. Initial program 56.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 66.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto t + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.9%

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t - x}{z}}, t\right) \]

   if -7.99999999999999996e89 < z < 1.04999999999999994e-5

   1. Initial program 92.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 94.2

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 71.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Applied rewrites 71.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   if 1.04999999999999994e-5 < z

   1. Initial program 77.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 78.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 37.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.5%

       \[\leadsto \frac{z}{z - a} \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e+62)
   (fma a (/ (- t x) z) t)
   (if (<= z 1.05e-5) (fma y (/ (- t x) a) x) (* t (/ z (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e+62) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 1.05e-5) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t * (z / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e+62)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 1.05e-5)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(t * Float64(z / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e+62], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.05e-5], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.19999999999999968e62

   1. Initial program 56.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{y \cdot \left(t - x\right)}{z}\right) - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{t + \left(-1 \cdot \frac{y \cdot \left(t - x\right)}{z} - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto t + \color{blue}{-1 \cdot \left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)} \]

      3. div-sub N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      4. associate-*r/ N/A

         \[\leadsto t + \color{blue}{\frac{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}{z}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) - -1 \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}}{z} \]

      7. associate-*r* N/A

         \[\leadsto t + \frac{\color{blue}{\left(-1 \cdot y\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      8. metadata-eval N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{1} \cdot \left(a \cdot \left(t - x\right)\right)}{z} \]

      9. *-lft-identity N/A

         \[\leadsto t + \frac{\left(-1 \cdot y\right) \cdot \left(t - x\right) + \color{blue}{a \cdot \left(t - x\right)}}{z} \]

      10. distribute-rgt-in N/A

          \[\leadsto t + \frac{\color{blue}{\left(t - x\right) \cdot \left(-1 \cdot y + a\right)}}{z} \]

      11. +-commutative N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a + -1 \cdot y\right)}}{z} \]

      12. mul-1-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \left(a + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}{z} \]

      13. sub-neg N/A

          \[\leadsto t + \frac{\left(t - x\right) \cdot \color{blue}{\left(a - y\right)}}{z} \]

      14. *-commutative N/A

          \[\leadsto t + \frac{\color{blue}{\left(a - y\right) \cdot \left(t - x\right)}}{z} \]

   7. Applied rewrites 78.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a - y, \frac{t - x}{z}, t\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto t + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 59.6%

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t - x}{z}}, t\right) \]

   if -5.19999999999999968e62 < z < 1.04999999999999994e-5

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t - x}{a}}, x\right) \]

      5. lower--.f64 72.2

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

   if 1.04999999999999994e-5 < z

   1. Initial program 77.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 78.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 37.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.5%

       \[\leadsto \frac{z}{z - a} \cdot t \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ z (- z a)))))
   (if (<= z -5.2e+62) t_1 (if (<= z 1.05e-5) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -5.2e+62) {
		tmp = t_1;
	} else if (z <= 1.05e-5) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z / Float64(z - a)))
	tmp = 0.0
	if (z <= -5.2e+62)
		tmp = t_1;
	elseif (z <= 1.05e-5)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+62], t$95$1, If[LessEqual[z, 1.05e-5], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999968e62 or 1.04999999999999994e-5 < z

   1. Initial program 67.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 40.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 57.5%

       \[\leadsto \frac{z}{z - a} \cdot t \]

   if -5.19999999999999968e62 < z < 1.04999999999999994e-5

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(t - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{t - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{t - x}{a}}, x\right) \]

      5. lower--.f64 72.2

         \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{t - x}}{a}, x\right) \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ z (- z a)))))
   (if (<= z -5e+62) t_1 (if (<= z 4.7e-6) (+ x (/ (* y t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -5e+62) {
		tmp = t_1;
	} else if (z <= 4.7e-6) {
		tmp = x + ((y * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z / (z - a))
    if (z <= (-5d+62)) then
        tmp = t_1
    else if (z <= 4.7d-6) then
        tmp = x + ((y * t) / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -5e+62) {
		tmp = t_1;
	} else if (z <= 4.7e-6) {
		tmp = x + ((y * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * (z / (z - a))
	tmp = 0
	if z <= -5e+62:
		tmp = t_1
	elif z <= 4.7e-6:
		tmp = x + ((y * t) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z / Float64(z - a)))
	tmp = 0.0
	if (z <= -5e+62)
		tmp = t_1;
	elseif (z <= 4.7e-6)
		tmp = Float64(x + Float64(Float64(y * t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z / (z - a));
	tmp = 0.0;
	if (z <= -5e+62)
		tmp = t_1;
	elseif (z <= 4.7e-6)
		tmp = x + ((y * t) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+62], t$95$1, If[LessEqual[z, 4.7e-6], N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.00000000000000029e62 or 4.69999999999999989e-6 < z

   1. Initial program 67.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 40.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 57.5%

       \[\leadsto \frac{z}{z - a} \cdot t \]

   if -5.00000000000000029e62 < z < 4.69999999999999989e-6

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(t - x\right)}}{a} \]

      3. lower--.f64 68.7

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(t - x\right)}}{a} \]

   5. Applied rewrites 68.7%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]

   6. Taylor expanded in t around inf

      \[\leadsto x + \frac{t \cdot y}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.9%

      \[\leadsto x + \frac{t \cdot y}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{z}{z - a}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ z (- z a)))))
   (if (<= z -7.2e+60) t_1 (if (<= z 2.05e-7) (* t (/ y (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -7.2e+60) {
		tmp = t_1;
	} else if (z <= 2.05e-7) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (z / (z - a))
    if (z <= (-7.2d+60)) then
        tmp = t_1
    else if (z <= 2.05d-7) then
        tmp = t * (y / (a - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (z / (z - a));
	double tmp;
	if (z <= -7.2e+60) {
		tmp = t_1;
	} else if (z <= 2.05e-7) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * (z / (z - a))
	tmp = 0
	if z <= -7.2e+60:
		tmp = t_1
	elif z <= 2.05e-7:
		tmp = t * (y / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(z / Float64(z - a)))
	tmp = 0.0
	if (z <= -7.2e+60)
		tmp = t_1;
	elseif (z <= 2.05e-7)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (z / (z - a));
	tmp = 0.0;
	if (z <= -7.2e+60)
		tmp = t_1;
	elseif (z <= 2.05e-7)
		tmp = t * (y / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+60], t$95$1, If[LessEqual[z, 2.05e-7], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.19999999999999935e60 or 2.05e-7 < z

   1. Initial program 67.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 73.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 40.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 57.5%

       \[\leadsto \frac{z}{z - a} \cdot t \]

   if -7.19999999999999935e60 < z < 2.05e-7

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 43.2

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{t \cdot y}{\color{blue}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 32.2%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+60}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 38.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-6}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 + \frac{a}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+61)
   (fma a (/ t z) t)
   (if (<= z 9.6e-6) (* t (/ y (- a z))) (* t (+ 1.0 (/ a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+61) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 9.6e-6) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t * (1.0 + (a / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+61)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 9.6e-6)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	else
		tmp = Float64(t * Float64(1.0 + Float64(a / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+61], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 9.6e-6], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(1.0 + N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.9999999999999996e61

   1. Initial program 56.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 67.4

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 42.8

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 42.8%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.6%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t + \frac{a \cdot t}{\color{blue}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 49.8%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]

   if -7.9999999999999996e61 < z < 9.5999999999999996e-6

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 95.3

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 43.2

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 43.2%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{t \cdot y}{\color{blue}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 32.2%

       \[\leadsto t \cdot \color{blue}{\frac{y}{a - z}} \]

   if 9.5999999999999996e-6 < z

   1. Initial program 77.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 78.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 37.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t \cdot \left(1 + \frac{a}{\color{blue}{z}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 48.3%

       \[\leadsto t \cdot \left(1 + \frac{a}{\color{blue}{z}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 + \frac{a}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.9e-64)
   (fma a (/ t z) t)
   (if (<= z 4.2e-6) (* y (/ t a)) (* t (+ 1.0 (/ a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e-64) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 4.2e-6) {
		tmp = y * (t / a);
	} else {
		tmp = t * (1.0 + (a / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e-64)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 4.2e-6)
		tmp = Float64(y * Float64(t / a));
	else
		tmp = Float64(t * Float64(1.0 + Float64(a / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e-64], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 4.2e-6], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(t * N[(1.0 + N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8999999999999997e-64

   1. Initial program 70.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 76.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 44.4

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.2%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t + \frac{a \cdot t}{\color{blue}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 37.6%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]

   if -3.8999999999999997e-64 < z < 4.1999999999999996e-6

   1. Initial program 95.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 42.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.7%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.3%

       \[\leadsto \frac{t}{a} \cdot y \]

   if 4.1999999999999996e-6 < z

   1. Initial program 77.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 78.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 37.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t \cdot \left(1 + \frac{a}{\color{blue}{z}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 48.3%

       \[\leadsto t \cdot \left(1 + \frac{a}{\color{blue}{z}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 + \frac{a}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.9e-64)
   (fma a (/ t z) t)
   (if (<= z 4.2e-6) (* y (/ t a)) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e-64) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 4.2e-6) {
		tmp = y * (t / a);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e-64)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 4.2e-6)
		tmp = Float64(y * Float64(t / a));
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e-64], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 4.2e-6], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8999999999999997e-64

   1. Initial program 70.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 76.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 44.4

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.2%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t + \frac{a \cdot t}{\color{blue}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 37.6%

       \[\leadsto \mathsf{fma}\left(a, \frac{t}{\color{blue}{z}}, t\right) \]

   if -3.8999999999999997e-64 < z < 4.1999999999999996e-6

   1. Initial program 95.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 42.0

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.7%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.3%

       \[\leadsto \frac{t}{a} \cdot y \]

   if 4.1999999999999996e-6 < z

   1. Initial program 77.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 78.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 37.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 56.5%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 48.3%

       \[\leadsto t \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-54}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.5e-54) (* t 1.0) (if (<= z 4.2e-6) (* y (/ t a)) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e-54) {
		tmp = t * 1.0;
	} else if (z <= 4.2e-6) {
		tmp = y * (t / a);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-9.5d-54)) then
        tmp = t * 1.0d0
    else if (z <= 4.2d-6) then
        tmp = y * (t / a)
    else
        tmp = t * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e-54) {
		tmp = t * 1.0;
	} else if (z <= 4.2e-6) {
		tmp = y * (t / a);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.5e-54:
		tmp = t * 1.0
	elif z <= 4.2e-6:
		tmp = y * (t / a)
	else:
		tmp = t * 1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.5e-54)
		tmp = Float64(t * 1.0);
	elseif (z <= 4.2e-6)
		tmp = Float64(y * Float64(t / a));
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.5e-54)
		tmp = t * 1.0;
	elseif (z <= 4.2e-6)
		tmp = y * (t / a);
	else
		tmp = t * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.5e-54], N[(t * 1.0), $MachinePrecision], If[LessEqual[z, 4.2e-6], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-54 or 4.1999999999999996e-6 < z

   1. Initial program 72.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 77.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 42.2

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 42.2%

       \[\leadsto t \cdot 1 \]

   if -9.4999999999999994e-54 < z < 4.1999999999999996e-6

   1. Initial program 95.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 41.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.3%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.8%

       \[\leadsto \frac{t}{a} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-54}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 35.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-54}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.5e-54) (* t 1.0) (if (<= z 4.2e-6) (* t (/ y a)) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e-54) {
		tmp = t * 1.0;
	} else if (z <= 4.2e-6) {
		tmp = t * (y / a);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-9.5d-54)) then
        tmp = t * 1.0d0
    else if (z <= 4.2d-6) then
        tmp = t * (y / a)
    else
        tmp = t * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e-54) {
		tmp = t * 1.0;
	} else if (z <= 4.2e-6) {
		tmp = t * (y / a);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.5e-54:
		tmp = t * 1.0
	elif z <= 4.2e-6:
		tmp = t * (y / a)
	else:
		tmp = t * 1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.5e-54)
		tmp = Float64(t * 1.0);
	elseif (z <= 4.2e-6)
		tmp = Float64(t * Float64(y / a));
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.5e-54)
		tmp = t * 1.0;
	elseif (z <= 4.2e-6)
		tmp = t * (y / a);
	else
		tmp = t * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.5e-54], N[(t * 1.0), $MachinePrecision], If[LessEqual[z, 4.2e-6], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-54 or 4.1999999999999996e-6 < z

   1. Initial program 72.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 77.1

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 42.2

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

   11. Taylor expanded in z around inf

       \[\leadsto t \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 42.2%

       \[\leadsto t \cdot 1 \]

   if -9.4999999999999994e-54 < z < 4.1999999999999996e-6

   1. Initial program 95.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

      5. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      7. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

      10. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

      11. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

      12. clear-num N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

      13. flip-- N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      16. lower-/.f64 96.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

   4. Applied rewrites 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   6. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

      6. lower--.f64 41.3

         \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

   7. Applied rewrites 41.3%

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto \frac{t \cdot y}{\color{blue}{a}} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.3%

       \[\leadsto \frac{y \cdot t}{\color{blue}{a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.0%

       \[\leadsto t \cdot \frac{y}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 25.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ t \cdot 1 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* t 1.0))

C

double code(double x, double y, double z, double t, double a) {
	return t * 1.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = t * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return t * 1.0;
}

Python

def code(x, y, z, t, a):
	return t * 1.0

Julia

function code(x, y, z, t, a)
	return Float64(t * 1.0)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t * 1.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(t * 1.0), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + x \]

   4. lift-/.f64 N/A

      \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t - x}{a - z}} + x \]

   5. clear-num N/A

      \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

   7. div-inv N/A

      \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

   9. lift--.f64 N/A

      \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{t - x}}} + x \]

   10. flip-- N/A

       \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{t \cdot t - x \cdot x}{t + x}}}} + x \]

   11. clear-num N/A

       \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t + x}{t \cdot t - x \cdot x}}} + x \]

   12. clear-num N/A

       \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{t \cdot t - x \cdot x}{t + x}} + x \]

   13. flip-- N/A

       \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

   14. lift--.f64 N/A

       \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

   15. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   16. lower-/.f64 85.9

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

4. Applied rewrites 85.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

5. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
6. Step-by-step derivation

   1. div-sub N/A

      \[\leadsto t \cdot \color{blue}{\frac{y - z}{a - z}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{t \cdot \left(y - z\right)}}{a - z} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{t \cdot \color{blue}{\left(y - z\right)}}{a - z} \]

   6. lower--.f64 41.8

      \[\leadsto \frac{t \cdot \left(y - z\right)}{\color{blue}{a - z}} \]

7. Applied rewrites 41.8%

   \[\leadsto \color{blue}{\frac{t \cdot \left(y - z\right)}{a - z}} \]

8. Taylor expanded in y around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
9. Step-by-step derivation

10. Applied rewrites 31.8%

    \[\leadsto t \cdot \color{blue}{\frac{z}{z + \left(-a\right)}} \]

11. Taylor expanded in z around inf

    \[\leadsto t \cdot 1 \]
12. Step-by-step derivation

13. Applied rewrites 25.2%

    \[\leadsto t \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))