Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.8% → 95.0%

Time: 10.3s

Alternatives: 18

Speedup: 0.3×

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: 79.8% 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: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{t - x}{\frac{-\left(a - z\right)}{y - z}}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_1 -1e-307)
     (fma (/ (- y z) (- a z)) (- t x) x)
     (if (<= t_1 0.0)
       (fma (fma -1.0 t x) (/ (- y a) z) t)
       (fma -1.0 (/ (- t x) (/ (- (- a z)) (- y z))) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_1 <= -1e-307) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else if (t_1 <= 0.0) {
		tmp = fma(fma(-1.0, t, x), ((y - a) / z), t);
	} else {
		tmp = fma(-1.0, ((t - x) / (-(a - z) / (y - z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_1 <= -1e-307)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x);
	elseif (t_1 <= 0.0)
		tmp = fma(fma(-1.0, t, x), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(-1.0, Float64(Float64(t - x) / Float64(Float64(-Float64(a - z)) / Float64(y - z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-307], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(-1.0 * t + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(-1.0 * N[(N[(t - x), $MachinePrecision] / N[((-N[(a - z), $MachinePrecision]) / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999909e-308

   1. Initial program 86.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.2

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

   4. Applied rewrites 95.2%

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

   if -9.99999999999999909e-308 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 2.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

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

   1. Initial program 89.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. *-lft-identity N/A

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

      7. remove-double-div N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/r/ N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 96.6%

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

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-307} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -1e-307) (not (<= t_1 0.0)))
     (fma (/ (- y z) (- a z)) (- t x) x)
     (fma (fma -1.0 t x) (/ (- y a) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0)) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else {
		tmp = fma(fma(-1.0, t, x), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -1e-307) || !(t_1 <= 0.0))
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x);
	else
		tmp = fma(fma(-1.0, t, x), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-307], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 * t + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999909e-308 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 87.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.8

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

   4. Applied rewrites 95.8%

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

   if -9.99999999999999909e-308 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 2.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 96.6%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-307} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-200} \lor \neg \left(t\_1 \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -4e-200) (not (<= t_1 2e-282)))
     (fma (/ (- x t) (- z a)) (- y z) x)
     (fma (fma -1.0 t x) (/ (- y a) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if ((t_1 <= -4e-200) || !(t_1 <= 2e-282)) {
		tmp = fma(((x - t) / (z - a)), (y - z), x);
	} else {
		tmp = fma(fma(-1.0, t, x), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -4e-200) || !(t_1 <= 2e-282))
		tmp = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x);
	else
		tmp = fma(fma(-1.0, t, x), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e-200], N[Not[LessEqual[t$95$1, 2e-282]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(N[(-1.0 * t + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -3.9999999999999999e-200 or 2e-282 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 91.8

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{0 - \left(a - z\right)}}, y - z, x\right) \]

      18. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a - z\right)}}, y - z, x\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}, y - z, x\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}, y - z, x\right) \]

      21. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}, y - z, x\right) \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if -3.9999999999999999e-200 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2e-282

   1. Initial program 10.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 86.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -4 \cdot 10^{-200} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 2 \cdot 10^{-282}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.32e+39) (not (<= z 7.8e+28)))
   (fma (fma -1.0 t x) (/ (- y a) z) t)
   (fma (/ (- y z) a) (- t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.32e+39) || !(z <= 7.8e+28)) {
		tmp = fma(fma(-1.0, t, x), ((y - a) / z), t);
	} else {
		tmp = fma(((y - z) / a), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.32e+39) || !(z <= 7.8e+28))
		tmp = fma(fma(-1.0, t, x), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.32e+39], N[Not[LessEqual[z, 7.8e+28]], $MachinePrecision]], N[(N[(-1.0 * t + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e39 or 7.7999999999999997e28 < z

   1. Initial program 61.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.4%

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

   if -1.32e39 < z < 7.7999999999999997e28

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 82.9

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

   7. Applied rewrites 82.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, t, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y \cdot t}{z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (/ (* y t) z))))
   (if (<= z -1.85e+92)
     t_1
     (if (<= z -7.2e+42)
       (* (/ (- x t) z) y)
       (if (<= z 6.4e+19) (fma y (/ t a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y * t) / z);
	double tmp;
	if (z <= -1.85e+92) {
		tmp = t_1;
	} else if (z <= -7.2e+42) {
		tmp = ((x - t) / z) * y;
	} else if (z <= 6.4e+19) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y * t) / z))
	tmp = 0.0
	if (z <= -1.85e+92)
		tmp = t_1;
	elseif (z <= -7.2e+42)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (z <= 6.4e+19)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.85e+92], t$95$1, If[LessEqual[z, -7.2e+42], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 6.4e+19], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999999e92 or 6.4e19 < z

   1. Initial program 60.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.4%

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

   if -1.84999999999999999e92 < z < -7.2000000000000002e42

   1. Initial program 67.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 68.0%

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

   if -7.2000000000000002e42 < z < 6.4e19

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+92}:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+39} \lor \neg \left(z \leq 1.15 \cdot 10^{+29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.3e+39) (not (<= z 1.15e+29)))
   (fma (/ (- x t) z) y t)
   (fma (/ (- y z) a) (- t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.3e+39) || !(z <= 1.15e+29)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma(((y - z) / a), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.3e+39) || !(z <= 1.15e+29))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.3e+39], N[Not[LessEqual[z, 1.15e+29]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.29999999999999979e39 or 1.1500000000000001e29 < z

   1. Initial program 61.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.9%

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

   if -5.29999999999999979e39 < z < 1.1500000000000001e29

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 82.9

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

   7. Applied rewrites 82.9%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+39} \lor \neg \left(z \leq 1.15 \cdot 10^{+29}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.2e+39) (not (<= z 7.8e+28)))
   (fma (/ (- x t) z) y t)
   (fma (- y z) (/ (- t x) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.2e+39) || !(z <= 7.8e+28)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma((y - z), ((t - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.2e+39) || !(z <= 7.8e+28))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.2e+39], N[Not[LessEqual[z, 7.8e+28]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e39 or 7.7999999999999997e28 < z

   1. Initial program 61.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.9%

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

   if -5.2e39 < z < 7.7999999999999997e28

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 80.0

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

   5. Applied rewrites 80.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+39} \lor \neg \left(z \leq 7.8 \cdot 10^{+28}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-5} \lor \neg \left(z \leq 7.6 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.9e-5) (not (<= z 7.6e+17)))
   (fma (/ (- x t) z) y t)
   (fma (/ y a) (- t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.9e-5) || !(z <= 7.6e+17)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma((y / a), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.9e-5) || !(z <= 7.6e+17))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(y / a), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.9e-5], N[Not[LessEqual[z, 7.6e+17]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000001e-5 or 7.6e17 < z

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 67.9%

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

   if -1.9000000000000001e-5 < z < 7.6e17

   1. Initial program 89.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

   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 79.9

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

   7. Applied rewrites 79.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-5} \lor \neg \left(z \leq 7.6 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+39} \lor \neg \left(z \leq 7.6 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.8e+39) (not (<= z 7.6e+17)))
   (fma (/ (- x t) z) y t)
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.8e+39) || !(z <= 7.6e+17)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.8e+39) || !(z <= 7.6e+17))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.8e+39], N[Not[LessEqual[z, 7.6e+17]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000002e39 or 7.6e17 < z

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.4%

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

   if -4.8000000000000002e39 < z < 7.6e17

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.3

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

   5. Applied rewrites 75.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+39} \lor \neg \left(z \leq 7.6 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+39} \lor \neg \left(z \leq 5.4 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.6e+39) (not (<= z 5.4e+17)))
   (fma (/ (- x t) z) y t)
   (fma y (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.6e+39) || !(z <= 5.4e+17)) {
		tmp = fma(((x - t) / z), y, t);
	} else {
		tmp = fma(y, (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.6e+39) || !(z <= 5.4e+17))
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	else
		tmp = fma(y, Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.6e+39], N[Not[LessEqual[z, 5.4e+17]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.60000000000000024e39 or 5.4e17 < z

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      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. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.4%

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

   if -4.60000000000000024e39 < z < 5.4e17

   1. Initial program 90.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.6

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+39} \lor \neg \left(z \leq 5.4 \cdot 10^{+17}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+31} \lor \neg \left(a \leq 29500000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7.5e+31) (not (<= a 29500000.0)))
   (fma y (/ t a) x)
   (- t (/ (* y t) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7.5e+31) || !(a <= 29500000.0)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t - ((y * t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7.5e+31) || !(a <= 29500000.0))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(t - Float64(Float64(y * t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7.5e+31], N[Not[LessEqual[a, 29500000.0]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e31 or 2.95e7 < a

   1. Initial program 89.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 92.0

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

   4. Applied rewrites 92.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 67.3

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.7%

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

   if -7.5e31 < a < 2.95e7

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around 0

      \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.5%

       \[\leadsto t - \frac{y \cdot t}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+31} \lor \neg \left(a \leq 29500000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 29500000:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.5e+31)
   (fma y (/ t a) x)
   (if (<= a 29500000.0) (- t (/ (* y t) z)) (+ x (/ (* t y) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.5e+31) {
		tmp = fma(y, (t / a), x);
	} else if (a <= 29500000.0) {
		tmp = t - ((y * t) / z);
	} else {
		tmp = x + ((t * y) / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.5e+31)
		tmp = fma(y, Float64(t / a), x);
	elseif (a <= 29500000.0)
		tmp = Float64(t - Float64(Float64(y * t) / z));
	else
		tmp = Float64(x + Float64(Float64(t * y) / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.5e+31], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 29500000.0], N[(t - N[(N[(y * t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.5e31

   1. Initial program 86.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 66.4

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 59.3%

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

   if -7.5e31 < a < 2.95e7

   1. Initial program 67.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.7%

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

   9. Taylor expanded in y around 0

      \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.5%

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

   if 2.95e7 < a

   1. Initial program 91.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto x + \frac{t \cdot y}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.8%

      \[\leadsto x + \frac{t \cdot y}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 29500000:\\ \;\;\;\;t - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+116} \lor \neg \left(z \leq 1.22 \cdot 10^{+20}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.35e+116) (not (<= z 1.22e+20)))
   (* (- t) -1.0)
   (fma y (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.35e+116) || !(z <= 1.22e+20)) {
		tmp = -t * -1.0;
	} else {
		tmp = fma(y, (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.35e+116) || !(z <= 1.22e+20))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = fma(y, Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.35e+116], N[Not[LessEqual[z, 1.22e+20]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e116 or 1.22e20 < z

   1. Initial program 59.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 54.7

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

   5. Applied rewrites 54.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 59.9%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 52.7%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -1.35e116 < z < 1.22e20

   1. Initial program 88.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 93.2

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 70.8

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+116} \lor \neg \left(z \leq 1.22 \cdot 10^{+20}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+90} \lor \neg \left(z \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.2e+90) (not (<= z 1.3e+19))) (* (- t) -1.0) (* (/ y a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.2e+90) || !(z <= 1.3e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = (y / a) * t;
	}
	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 <= (-3.2d+90)) .or. (.not. (z <= 1.3d+19))) then
        tmp = -t * (-1.0d0)
    else
        tmp = (y / a) * t
    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 <= -3.2e+90) || !(z <= 1.3e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = (y / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.2e+90) or not (z <= 1.3e+19):
		tmp = -t * -1.0
	else:
		tmp = (y / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.2e+90) || !(z <= 1.3e+19))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = Float64(Float64(y / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.2e+90) || ~((z <= 1.3e+19)))
		tmp = -t * -1.0;
	else
		tmp = (y / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.2e+90], N[Not[LessEqual[z, 1.3e+19]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999998e90 or 1.3e19 < z

   1. Initial program 60.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 51.8%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -3.19999999999999998e90 < z < 1.3e19

   1. Initial program 88.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 31.0%

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

   12. Applied rewrites 34.3%

       \[\leadsto \frac{y}{a} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+90} \lor \neg \left(z \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+90} \lor \neg \left(z \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.2e+90) (not (<= z 1.3e+19))) (* (- t) -1.0) (* y (/ t a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.2e+90) || !(z <= 1.3e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = y * (t / a);
	}
	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 <= (-3.2d+90)) .or. (.not. (z <= 1.3d+19))) then
        tmp = -t * (-1.0d0)
    else
        tmp = y * (t / a)
    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 <= -3.2e+90) || !(z <= 1.3e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = y * (t / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.2e+90) or not (z <= 1.3e+19):
		tmp = -t * -1.0
	else:
		tmp = y * (t / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.2e+90) || !(z <= 1.3e+19))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = Float64(y * Float64(t / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.2e+90) || ~((z <= 1.3e+19)))
		tmp = -t * -1.0;
	else
		tmp = y * (t / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.2e+90], N[Not[LessEqual[z, 1.3e+19]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999998e90 or 1.3e19 < z

   1. Initial program 60.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 51.8%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -3.19999999999999998e90 < z < 1.3e19

   1. Initial program 88.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 31.0%

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

   12. Applied rewrites 33.4%

       \[\leadsto y \cdot \frac{t}{\color{blue}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+90} \lor \neg \left(z \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \lor \neg \left(z \leq 6.8 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.0) (not (<= z 6.8e+19))) (* (- t) -1.0) (* 1.0 x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.0) || !(z <= 6.8e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-4.0d0)) .or. (.not. (z <= 6.8d+19))) then
        tmp = -t * (-1.0d0)
    else
        tmp = 1.0d0 * x
    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 <= -4.0) || !(z <= 6.8e+19)) {
		tmp = -t * -1.0;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.0) or not (z <= 6.8e+19):
		tmp = -t * -1.0
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.0) || !(z <= 6.8e+19))
		tmp = Float64(Float64(-t) * -1.0);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.0) || ~((z <= 6.8e+19)))
		tmp = -t * -1.0;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.0], N[Not[LessEqual[z, 6.8e+19]], $MachinePrecision]], N[((-t) * -1.0), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4 or 6.8e19 < z

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 56.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(-t\right) \cdot -1 \]
   10. Step-by-step derivation

   11. Applied rewrites 45.3%

       \[\leadsto \left(-t\right) \cdot -1 \]

   if -4 < z < 6.8e19

   1. Initial program 89.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 76.2

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 52.1%

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

   11. Taylor expanded in y around 0

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 33.0%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \lor \neg \left(z \leq 6.8 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-t\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+180} \lor \neg \left(z \leq 1.22 \cdot 10^{+20}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.5e+180) (not (<= z 1.22e+20))) (+ x (- t x)) (* 1.0 x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.5e+180) || !(z <= 1.22e+20)) {
		tmp = x + (t - x);
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-2.5d+180)) .or. (.not. (z <= 1.22d+20))) then
        tmp = x + (t - x)
    else
        tmp = 1.0d0 * x
    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 <= -2.5e+180) || !(z <= 1.22e+20)) {
		tmp = x + (t - x);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.5e+180) or not (z <= 1.22e+20):
		tmp = x + (t - x)
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.5e+180) || !(z <= 1.22e+20))
		tmp = Float64(x + Float64(t - x));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.5e+180) || ~((z <= 1.22e+20)))
		tmp = x + (t - x);
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.5e+180], N[Not[LessEqual[z, 1.22e+20]], $MachinePrecision]], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4999999999999998e180 or 1.22e20 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 39.4

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

   5. Applied rewrites 39.4%

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

   if -2.4999999999999998e180 < z < 1.22e20

   1. Initial program 87.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   6. 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 68.2

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 46.1%

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

   11. Taylor expanded in y around 0

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 30.4%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+180} \lor \neg \left(z \leq 1.22 \cdot 10^{+20}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

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 = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * x;
}

Python

def code(x, y, z, t, a):
	return 1.0 * x

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 77.8%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 85.0

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

4. Applied rewrites 85.0%

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

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
6. 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 50.2

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

7. Applied rewrites 50.2%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 34.9%

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

11. Taylor expanded in y around 0

    \[\leadsto 1 \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 23.5%

    \[\leadsto 1 \cdot x \]

14. Final simplification 23.5%

    \[\leadsto 1 \cdot x \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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)))))