Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.7% → 93.2%

Time: 15.7s

Alternatives: 21

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% 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: 93.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-232}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_1 -1e-232)
     t_1
     (if (<= t_1 0.0)
       (fma (/ (- t x) z) (- a y) t)
       (fma (/ (- y z) (- a z)) (- t x) 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-232) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = fma(((y - z) / (a - z)), (t - x), 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-232)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), 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-232], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $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)))) < -1.00000000000000002e-232

   1. Initial program 93.7%

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

   if -1.00000000000000002e-232 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.8%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 94.6%

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

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

   1. Initial program 90.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 95.7

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

   4. Applied egg-rr 95.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-232}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;x + y \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+237}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (<= t_2 -1e-252)
     (+ x (* y t_1))
     (if (<= t_2 5e-161)
       (+ t (/ (* (- t x) (- a y)) z))
       (if (<= t_2 4e+237)
         (+ x (* (- y z) (/ t (- a z))))
         (fma (/ y (- a z)) (- t x) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if (t_2 <= -1e-252) {
		tmp = x + (y * t_1);
	} else if (t_2 <= 5e-161) {
		tmp = t + (((t - x) * (a - y)) / z);
	} else if (t_2 <= 4e+237) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else {
		tmp = fma((y / (a - z)), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if (t_2 <= -1e-252)
		tmp = Float64(x + Float64(y * t_1));
	elseif (t_2 <= 5e-161)
		tmp = Float64(t + Float64(Float64(Float64(t - x) * Float64(a - y)) / z));
	elseif (t_2 <= 4e+237)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	else
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-252], N[(x + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-161], N[(t + N[(N[(N[(t - x), $MachinePrecision] * N[(a - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+237], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999943e-253

   1. Initial program 92.8%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 73.5%

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

   if -9.99999999999999943e-253 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.9999999999999999e-161

   1. Initial program 10.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 26.7

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

   4. Applied egg-rr 26.7%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 84.6%

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

   if 4.9999999999999999e-161 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 3.99999999999999976e237

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 89.0

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

   5. Simplified 89.0%

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

   if 3.99999999999999976e237 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 96.6

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 89.5

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

   7. Simplified 89.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-252}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t + \frac{\left(t - x\right) \cdot \left(a - y\right)}{z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 4 \cdot 10^{+237}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-232}:\\ \;\;\;\;x + y \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+237}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (<= t_2 -1e-232)
     (+ x (* y t_1))
     (if (<= t_2 5e-161)
       (+ t (/ (* y (- x t)) z))
       (if (<= t_2 4e+237)
         (+ x (* (- y z) (/ t (- a z))))
         (fma (/ y (- a z)) (- t x) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if (t_2 <= -1e-232) {
		tmp = x + (y * t_1);
	} else if (t_2 <= 5e-161) {
		tmp = t + ((y * (x - t)) / z);
	} else if (t_2 <= 4e+237) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else {
		tmp = fma((y / (a - z)), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if (t_2 <= -1e-232)
		tmp = Float64(x + Float64(y * t_1));
	elseif (t_2 <= 5e-161)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	elseif (t_2 <= 4e+237)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	else
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-232], N[(x + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-161], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+237], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000002e-232

   1. Initial program 93.7%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 74.2%

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

   if -1.00000000000000002e-232 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.9999999999999999e-161

   1. Initial program 10.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 28.3

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

   4. Applied egg-rr 28.3%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 82.8%

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

   8. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 72.3

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

   10. Simplified 72.3%

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

   if 4.9999999999999999e-161 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 3.99999999999999976e237

   1. Initial program 98.1%

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

   3. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 89.0

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

   5. Simplified 89.0%

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

   if 3.99999999999999976e237 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 96.6

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 89.5

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

   7. Simplified 89.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-232}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 5 \cdot 10^{-161}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 4 \cdot 10^{+237}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-252)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-252) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-252)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-252], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -9.99999999999999943e-253 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 94.1

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

   4. Applied egg-rr 94.1%

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

   if -9.99999999999999943e-253 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.8%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 97.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+80}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{a}, x\right)\\ \mathbf{elif}\;z \leq 8.5:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+80)
   t
   (if (<= z -7.8e+34)
     (* x (/ y z))
     (if (<= z 5.2e-96) (fma z (/ x a) x) (if (<= z 8.5) (* y (/ t a)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+80) {
		tmp = t;
	} else if (z <= -7.8e+34) {
		tmp = x * (y / z);
	} else if (z <= 5.2e-96) {
		tmp = fma(z, (x / a), x);
	} else if (z <= 8.5) {
		tmp = y * (t / a);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+80)
		tmp = t;
	elseif (z <= -7.8e+34)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= 5.2e-96)
		tmp = fma(z, Float64(x / a), x);
	elseif (z <= 8.5)
		tmp = Float64(y * Float64(t / a));
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+80], t, If[LessEqual[z, -7.8e+34], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-96], N[(z * N[(x / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 8.5], N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision], t]]]]

Derivation

1. Split input into 4 regimes

2. if z < -8e80 or 8.5 < z

   1. Initial program 63.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 50.3%

      \[\leadsto \color{blue}{t} \]

   if -8e80 < z < -7.80000000000000038e34

   1. Initial program 72.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. *-lft-identity N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. neg-lowering-neg.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. --lowering--.f64 56.1

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

   5. Simplified 56.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{a - z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in a around 0

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

       1. associate-/l* N/A

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

       2. *-lowering-*.f64 N/A

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

       3. /-lowering-/.f64 47.6

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

   11. Simplified 47.6%

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

   if -7.80000000000000038e34 < z < 5.2000000000000003e-96

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. *-lft-identity N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. neg-lowering-neg.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. --lowering--.f64 57.7

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

   5. Simplified 57.7%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 38.0

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

   8. Simplified 38.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{x \cdot z}{a}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

       5. /-lowering-/.f64 38.8

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

   11. Simplified 38.8%

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

   if 5.2000000000000003e-96 < z < 8.5

   1. Initial program 90.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 63.4

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

   5. Simplified 63.4%

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

   6. Taylor expanded in a around inf

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

   8. Simplified 47.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 56.2

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

   10. Applied egg-rr 56.2%

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

   11. Taylor expanded in y around inf

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

   13. Simplified 43.8%

       \[\leadsto \color{blue}{y} \cdot \frac{t}{a} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 35.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{t}{a}\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-277}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-99}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ t a))))
   (if (<= z -2.3e+44)
     t
     (if (<= z -1.52e-277) t_1 (if (<= z 2.5e-99) x (if (<= z 6.4) t_1 t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (t / a);
	double tmp;
	if (z <= -2.3e+44) {
		tmp = t;
	} else if (z <= -1.52e-277) {
		tmp = t_1;
	} else if (z <= 2.5e-99) {
		tmp = x;
	} else if (z <= 6.4) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = y * (t / a)
    if (z <= (-2.3d+44)) then
        tmp = t
    else if (z <= (-1.52d-277)) then
        tmp = t_1
    else if (z <= 2.5d-99) then
        tmp = x
    else if (z <= 6.4d0) then
        tmp = t_1
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (t / a);
	double tmp;
	if (z <= -2.3e+44) {
		tmp = t;
	} else if (z <= -1.52e-277) {
		tmp = t_1;
	} else if (z <= 2.5e-99) {
		tmp = x;
	} else if (z <= 6.4) {
		tmp = t_1;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (t / a)
	tmp = 0
	if z <= -2.3e+44:
		tmp = t
	elif z <= -1.52e-277:
		tmp = t_1
	elif z <= 2.5e-99:
		tmp = x
	elif z <= 6.4:
		tmp = t_1
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(t / a))
	tmp = 0.0
	if (z <= -2.3e+44)
		tmp = t;
	elseif (z <= -1.52e-277)
		tmp = t_1;
	elseif (z <= 2.5e-99)
		tmp = x;
	elseif (z <= 6.4)
		tmp = t_1;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (t / a);
	tmp = 0.0;
	if (z <= -2.3e+44)
		tmp = t;
	elseif (z <= -1.52e-277)
		tmp = t_1;
	elseif (z <= 2.5e-99)
		tmp = x;
	elseif (z <= 6.4)
		tmp = t_1;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.3e+44], t, If[LessEqual[z, -1.52e-277], t$95$1, If[LessEqual[z, 2.5e-99], x, If[LessEqual[z, 6.4], t$95$1, t]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.30000000000000004e44 or 6.4000000000000004 < z

   1. Initial program 64.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 47.2%

      \[\leadsto \color{blue}{t} \]

   if -2.30000000000000004e44 < z < -1.5199999999999999e-277 or 2.49999999999999985e-99 < z < 6.4000000000000004

   1. Initial program 94.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 44.9

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

   5. Simplified 44.9%

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

   6. Taylor expanded in a around inf

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

   8. Simplified 32.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 39.9

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

   10. Applied egg-rr 39.9%

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

   11. Taylor expanded in y around inf

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

   13. Simplified 35.5%

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

   if -1.5199999999999999e-277 < z < 2.49999999999999985e-99

   1. Initial program 94.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 49.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+58}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) z) (- a y) t)))
   (if (<= z -5.6e+42)
     t_1
     (if (<= z 8e+58) (+ x (* y (/ (- t x) (- a z)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / z), (a - y), t);
	double tmp;
	if (z <= -5.6e+42) {
		tmp = t_1;
	} else if (z <= 8e+58) {
		tmp = x + (y * ((t - x) / (a - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / z), Float64(a - y), t)
	tmp = 0.0
	if (z <= -5.6e+42)
		tmp = t_1;
	elseif (z <= 8e+58)
		tmp = Float64(x + Float64(y * Float64(Float64(t - x) / Float64(a - z))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5.6e+42], t$95$1, If[LessEqual[z, 8e+58], N[(x + N[(y * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5999999999999999e42 or 7.99999999999999955e58 < z

   1. Initial program 60.8%

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

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 78.9%

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

   if -5.5999999999999999e42 < z < 7.99999999999999955e58

   1. Initial program 94.1%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 87.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+58}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+59}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.8e+78)
   (fma a (/ (- t x) z) t)
   (if (<= z 3.5e+59)
     (+ x (* y (/ (- t x) (- a z))))
     (+ t (/ (* y (- x t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.8e+78) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 3.5e+59) {
		tmp = x + (y * ((t - x) / (a - z)));
	} else {
		tmp = t + ((y * (x - t)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.8e+78)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 3.5e+59)
		tmp = Float64(x + Float64(y * Float64(Float64(t - x) / Float64(a - z))));
	else
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.8e+78], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 3.5e+59], N[(x + N[(y * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.80000000000000034e78

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 65.5

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 68.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 64.3

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

   10. Simplified 64.3%

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

   if -5.80000000000000034e78 < z < 3.5e59

   1. Initial program 92.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 84.6%

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

   if 3.5e59 < z

   1. Initial program 64.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 69.1

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

   4. Applied egg-rr 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 74.8%

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

   8. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 72.2

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

   10. Simplified 72.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+59}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+79)
   (fma a (/ (- t x) z) t)
   (if (<= z 3.35e+59)
     (fma (/ y (- a z)) (- t x) x)
     (+ t (/ (* y (- x t)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+79) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 3.35e+59) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t + ((y * (x - t)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+79)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 3.35e+59)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	else
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+79], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 3.35e+59], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.70000000000000016e79

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 65.5

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 68.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 64.3

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

   10. Simplified 64.3%

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

   if -1.70000000000000016e79 < z < 3.3500000000000002e59

   1. Initial program 92.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 92.2

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

   4. Applied egg-rr 92.2%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 83.7

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

   7. Simplified 83.7%

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

   if 3.3500000000000002e59 < z

   1. Initial program 64.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 69.1

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

   4. Applied egg-rr 69.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 74.8%

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

   8. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 72.2

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

   10. Simplified 72.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -7 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 450:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -7e-19) t_1 (if (<= a 450.0) (+ t (/ (* y (- x t)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / a), x);
	double tmp;
	if (a <= -7e-19) {
		tmp = t_1;
	} else if (a <= 450.0) {
		tmp = t + ((y * (x - t)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -7e-19)
		tmp = t_1;
	elseif (a <= 450.0)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7e-19], t$95$1, If[LessEqual[a, 450.0], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.00000000000000031e-19 or 450 < a

   1. Initial program 87.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 76.7

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

   5. Simplified 76.7%

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

   if -7.00000000000000031e-19 < a < 450

   1. Initial program 72.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 74.7

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

   4. Applied egg-rr 74.7%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 73.4%

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

   8. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 70.3

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

   10. Simplified 70.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 450:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 260000:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- t x) x)))
   (if (<= a -5e-21) t_1 (if (<= a 260000.0) (+ t (/ (* y (- x t)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), (t - x), x);
	double tmp;
	if (a <= -5e-21) {
		tmp = t_1;
	} else if (a <= 260000.0) {
		tmp = t + ((y * (x - t)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -5e-21)
		tmp = t_1;
	elseif (a <= 260000.0)
		tmp = Float64(t + Float64(Float64(y * Float64(x - t)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5e-21], t$95$1, If[LessEqual[a, 260000.0], N[(t + N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999973e-21 or 2.6e5 < a

   1. Initial program 86.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 90.7

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

   4. Applied egg-rr 90.7%

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

   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. /-lowering-/.f64 71.9

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

   7. Simplified 71.9%

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

   if -4.99999999999999973e-21 < a < 2.6e5

   1. Initial program 73.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 74.5

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

   4. Applied egg-rr 74.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 73.9%

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

   8. Taylor expanded in a around 0

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

      1. --lowering--.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 70.7

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

   10. Simplified 70.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 260000:\\ \;\;\;\;t + \frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e+78)
   (fma a (/ (- t x) z) t)
   (if (<= z 2.5e-101) (fma (/ y a) (- t x) x) (* (- y z) (/ t (- a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e+78) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 2.5e-101) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = (y - z) * (t / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e+78)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 2.5e-101)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e+78], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 2.5e-101], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6000000000000004e78

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 65.5

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 68.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 64.3

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

   10. Simplified 64.3%

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

   if -4.6000000000000004e78 < z < 2.5e-101

   1. Initial program 92.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 93.1

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

   4. Applied egg-rr 93.1%

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

   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. /-lowering-/.f64 74.7

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

   7. Simplified 74.7%

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

   if 2.5e-101 < z

   1. Initial program 75.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 52.7

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

   5. Simplified 52.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 62.1

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

   7. Applied egg-rr 62.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.26e+79)
   (fma a (/ (- t x) z) t)
   (if (<= z 8e+83) (fma (/ y a) (- t x) x) (* t (/ z (- z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.26e+79) {
		tmp = fma(a, ((t - x) / z), t);
	} else if (z <= 8e+83) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t * (z / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.26e+79)
		tmp = fma(a, Float64(Float64(t - x) / z), t);
	elseif (z <= 8e+83)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(t * Float64(z / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.26e+79], N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 8e+83], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.26e79

   1. Initial program 57.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 65.5

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 68.8%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 64.3

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

   10. Simplified 64.3%

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

   if -1.26e79 < z < 8.00000000000000025e83

   1. Initial program 90.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 90.8

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 68.3

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

   7. Simplified 68.3%

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

   if 8.00000000000000025e83 < z

   1. Initial program 64.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 44.0

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

   5. Simplified 44.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 62.2

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

   8. Simplified 62.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -1.7e+78) t_1 (if (<= z 1.55e+83) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -1.7e+78) {
		tmp = t_1;
	} else if (z <= 1.55e+83) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -1.7e+78)
		tmp = t_1;
	elseif (z <= 1.55e+83)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.7e+78], t$95$1, If[LessEqual[z, 1.55e+83], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000004e78 or 1.54999999999999996e83 < z

   1. Initial program 60.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 67.1

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

   4. Applied egg-rr 67.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 72.0%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 62.5

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

   10. Simplified 62.5%

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

   if -1.70000000000000004e78 < z < 1.54999999999999996e83

   1. Initial program 91.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   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. /-lowering-/.f64 68.7

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

   7. Simplified 68.7%

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

Alternative 15: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -2e+78) t_1 (if (<= z 1.8e+83) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -2e+78) {
		tmp = t_1;
	} else if (z <= 1.8e+83) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -2e+78)
		tmp = t_1;
	elseif (z <= 1.8e+83)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -2e+78], t$95$1, If[LessEqual[z, 1.8e+83], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000002e78 or 1.7999999999999999e83 < z

   1. Initial program 60.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 67.1

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

   4. Applied egg-rr 67.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

   7. Simplified 72.0%

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

   8. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 62.5

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

   10. Simplified 62.5%

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

   if -2.00000000000000002e78 < z < 1.7999999999999999e83

   1. Initial program 91.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 68.4

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

   5. Simplified 68.4%

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

Alternative 16: 56.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{t - x}{z}, t\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- t x) z) t)))
   (if (<= z -1.3e+48) t_1 (if (<= z 1.7e+83) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((t - x) / z), t);
	double tmp;
	if (z <= -1.3e+48) {
		tmp = t_1;
	} else if (z <= 1.7e+83) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(t - x) / z), t)
	tmp = 0.0
	if (z <= -1.3e+48)
		tmp = t_1;
	elseif (z <= 1.7e+83)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.3e+48], t$95$1, If[LessEqual[z, 1.7e+83], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999998e48 or 1.6999999999999999e83 < z

   1. Initial program 60.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 67.1

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

   4. Applied egg-rr 67.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

         \[\leadsto t + \frac{\color{blue}{-1 \cdot \left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right)}}{z} \]

      8. associate-*r/ N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      9. mul-1-neg N/A

         \[\leadsto t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right)} \]

      10. unsub-neg N/A

          \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      11. --lowering--.f64 N/A

          \[\leadsto \color{blue}{t - \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto t - \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

   7. Simplified 71.2%

      \[\leadsto \color{blue}{t - \frac{\left(t - x\right) \cdot \left(y - a\right)}{z}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{t - -1 \cdot \frac{a \cdot \left(t - x\right)}{z}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{a \cdot \left(t - x\right)}{z}} \]

      2. metadata-eval N/A

         \[\leadsto t + \color{blue}{1} \cdot \frac{a \cdot \left(t - x\right)}{z} \]

      3. *-lft-identity N/A

         \[\leadsto t + \color{blue}{\frac{a \cdot \left(t - x\right)}{z}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot \left(t - x\right)}{z} + t} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{t - x}{z}} + t \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{t - x}{z}}, t\right) \]

      8. --lowering--.f64 58.8

         \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{t - x}}{z}, t\right) \]

   10. Simplified 58.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{t - x}{z}, t\right)} \]

   if -1.29999999999999998e48 < z < 1.6999999999999999e83

   1. Initial program 92.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      2. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      4. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, t - x, x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, t - x, x\right) \]

      14. --lowering--.f64 92.4

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{t - x}, x\right) \]

   4. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   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. /-lowering-/.f64 70.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Simplified 70.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 60.1%

       \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+78}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+78) t (if (<= z 1.6e+88) (fma (/ y a) t x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+78) {
		tmp = t;
	} else if (z <= 1.6e+88) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+78)
		tmp = t;
	elseif (z <= 1.6e+88)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+78], t, If[LessEqual[z, 1.6e+88], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e78 or 1.5999999999999999e88 < z

   1. Initial program 60.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 55.9%

      \[\leadsto \color{blue}{t} \]

   if -1.3e78 < z < 1.5999999999999999e88

   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. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      2. clear-num N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{1}{\frac{a - z}{t - x}}} + x \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 1}{\frac{a - z}{t - x}}} + x \]

      4. div-inv N/A

         \[\leadsto \frac{\left(y - z\right) \cdot 1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}} + x \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\color{blue}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}} + x \]

      9. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \color{blue}{\left(t - x\right)} + x \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - z}}{a - z}, t - x, x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{\color{blue}{a - z}}, t - x, x\right) \]

      14. --lowering--.f64 90.3

          \[\leadsto \mathsf{fma}\left(\frac{y - z}{a - z}, \color{blue}{t - x}, x\right) \]

   4. Applied egg-rr 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 68.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   7. Simplified 68.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
   9. Step-by-step derivation

   10. Simplified 57.6%

       \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+81}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.4e+81)
   t
   (if (<= z -7.5e+34) (* x (/ y z)) (if (<= z 3.1e+59) x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.4e+81) {
		tmp = t;
	} else if (z <= -7.5e+34) {
		tmp = x * (y / z);
	} else if (z <= 3.1e+59) {
		tmp = x;
	} else {
		tmp = 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 <= (-5.4d+81)) then
        tmp = t
    else if (z <= (-7.5d+34)) then
        tmp = x * (y / z)
    else if (z <= 3.1d+59) then
        tmp = x
    else
        tmp = 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 <= -5.4e+81) {
		tmp = t;
	} else if (z <= -7.5e+34) {
		tmp = x * (y / z);
	} else if (z <= 3.1e+59) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.4e+81:
		tmp = t
	elif z <= -7.5e+34:
		tmp = x * (y / z)
	elif z <= 3.1e+59:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.4e+81)
		tmp = t;
	elseif (z <= -7.5e+34)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= 3.1e+59)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.4e+81)
		tmp = t;
	elseif (z <= -7.5e+34)
		tmp = x * (y / z);
	elseif (z <= 3.1e+59)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.4e+81], t, If[LessEqual[z, -7.5e+34], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+59], x, t]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.3999999999999999e81 or 3.10000000000000015e59 < z

   1. Initial program 60.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 55.0%

      \[\leadsto \color{blue}{t} \]

   if -5.3999999999999999e81 < z < -7.49999999999999976e34

   1. Initial program 72.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - z}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(y - z\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - z}\right)\right) + \color{blue}{x} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \mathsf{neg}\left(\frac{x}{a - z}\right), x\right)} \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \mathsf{neg}\left(\frac{x}{a - z}\right), x\right) \]

      13. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\mathsf{neg}\left(\frac{x}{a - z}\right)}, x\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - z, \mathsf{neg}\left(\color{blue}{\frac{x}{a - z}}\right), x\right) \]

      15. --lowering--.f64 56.1

          \[\leadsto \mathsf{fma}\left(y - z, -\frac{x}{\color{blue}{a - z}}, x\right) \]

   5. Simplified 56.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, -\frac{x}{a - z}, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{a - z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot y}{a - z}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot y}{a - z}\right)} \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y}{a - z}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y}{a - z}}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \color{blue}{\frac{y}{a - z}}\right) \]

      6. --lowering--.f64 62.0

         \[\leadsto -x \cdot \frac{y}{\color{blue}{a - z}} \]

   8. Simplified 62.0%

      \[\leadsto \color{blue}{-x \cdot \frac{y}{a - z}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

       3. /-lowering-/.f64 47.6

          \[\leadsto x \cdot \color{blue}{\frac{y}{z}} \]

   11. Simplified 47.6%

       \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   if -7.49999999999999976e34 < z < 3.10000000000000015e59

   1. Initial program 94.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 31.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 38.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+71}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.75e+90) x (if (<= a 3e+71) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.75e+90) {
		tmp = x;
	} else if (a <= 3e+71) {
		tmp = t;
	} else {
		tmp = 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 (a <= (-1.75d+90)) then
        tmp = x
    else if (a <= 3d+71) then
        tmp = t
    else
        tmp = 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 (a <= -1.75e+90) {
		tmp = x;
	} else if (a <= 3e+71) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.75e+90:
		tmp = x
	elif a <= 3e+71:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.75e+90)
		tmp = x;
	elseif (a <= 3e+71)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.75e+90)
		tmp = x;
	elseif (a <= 3e+71)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.75e+90], x, If[LessEqual[a, 3e+71], t, x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7499999999999999e90 or 3.00000000000000013e71 < a

   1. Initial program 92.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 48.5%

      \[\leadsto \color{blue}{x} \]

   if -1.7499999999999999e90 < a < 3.00000000000000013e71

   1. Initial program 74.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Simplified 35.0%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.1% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 t)

C

double code(double x, double y, double z, double t, double a) {
	return t;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return t;
}

Python

def code(x, y, z, t, a):
	return t

Julia

function code(x, y, z, t, a)
	return t
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 80.1%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Simplified 26.0%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Alternative 21: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 80.1%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{y - z}{a - z}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{y - z}{a - z} + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y - z}{a - z}\right) \cdot x + 1 \cdot x} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z}\right)\right)} \cdot x + 1 \cdot x \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y - z}{a - z} \cdot x\right)\right)} + 1 \cdot x \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y - z}{a - z}}\right)\right) + 1 \cdot x \]

   6. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(y - z\right)}{a - z}}\right)\right) + 1 \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - z\right) \cdot x}}{a - z}\right)\right) + 1 \cdot x \]

   8. associate-/l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - z\right) \cdot \frac{x}{a - z}}\right)\right) + 1 \cdot x \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - z}\right)\right)} + 1 \cdot x \]

   10. *-lft-identity N/A

       \[\leadsto \left(y - z\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - z}\right)\right) + \color{blue}{x} \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \mathsf{neg}\left(\frac{x}{a - z}\right), x\right)} \]

   12. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y - z}, \mathsf{neg}\left(\frac{x}{a - z}\right), x\right) \]

   13. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{\mathsf{neg}\left(\frac{x}{a - z}\right)}, x\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - z, \mathsf{neg}\left(\color{blue}{\frac{x}{a - z}}\right), x\right) \]

   15. --lowering--.f64 39.6

       \[\leadsto \mathsf{fma}\left(y - z, -\frac{x}{\color{blue}{a - z}}, x\right) \]

5. Simplified 39.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, -\frac{x}{a - z}, x\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + -1 \cdot x} \]
7. Step-by-step derivation

   1. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 2.8

      \[\leadsto \color{blue}{0} \]

8. Simplified 2.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))