Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 81.1% → 94.9%

Time: 9.8s

Alternatives: 19

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.1% 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: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_2 -4e-302) t_1 (if (<= t_2 2e-275) (fma (/ x z) (- y a) 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 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_2 <= -4e-302) {
		tmp = t_1;
	} else if (t_2 <= 2e-275) {
		tmp = fma((x / z), (y - a), 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(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_2 <= -4e-302)
		tmp = t_1;
	elseif (t_2 <= 2e-275)
		tmp = fma(Float64(x / z), Float64(y - a), 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[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -4e-302], t$95$1, If[LessEqual[t$95$2, 2e-275], N[(N[(x / z), $MachinePrecision] * N[(y - a), $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)))) < -3.9999999999999999e-302 or 1.99999999999999987e-275 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 93.7

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

   4. Applied rewrites 93.7%

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

   if -3.9999999999999999e-302 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1.99999999999999987e-275

   1. Initial program 3.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -4 \cdot 10^{-302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 2 \cdot 10^{-275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_2 -1e-110)
     t_1
     (if (<= t_2 2e-275) (fma (/ (- x t) z) (- y a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double t_2 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_2 <= -1e-110) {
		tmp = t_1;
	} else if (t_2 <= 2e-275) {
		tmp = fma(((x - t) / z), (y - a), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	t_2 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_2 <= -1e-110)
		tmp = t_1;
	elseif (t_2 <= 2e-275)
		tmp = fma(Float64(Float64(x - t) / z), Float64(y - a), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-110], t$95$1, If[LessEqual[t$95$2, 2e-275], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $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)))) < -1.0000000000000001e-110 or 1.99999999999999987e-275 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 89.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.5

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 89.5

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

   4. Applied rewrites 89.5%

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

   if -1.0000000000000001e-110 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1.99999999999999987e-275

   1. Initial program 14.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 91.6%

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

   7. Applied rewrites 91.6%

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

   9. Applied rewrites 91.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 2 \cdot 10^{-275}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 20.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 10^{+286}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (* (/ (- t x) (- a z)) (- y z)) x) 1e+286)
   (+ (- t x) x)
   (* (/ t z) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((((t - x) / (a - z)) * (y - z)) + x) <= 1e+286) {
		tmp = (t - x) + x;
	} else {
		tmp = (t / z) * a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((((t - x) / (a - z)) * (y - z)) + x) <= 1d+286) then
        tmp = (t - x) + x
    else
        tmp = (t / z) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((((t - x) / (a - z)) * (y - z)) + x) <= 1e+286) {
		tmp = (t - x) + x;
	} else {
		tmp = (t / z) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((((t - x) / (a - z)) * (y - z)) + x) <= 1e+286:
		tmp = (t - x) + x
	else:
		tmp = (t / z) * a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x) <= 1e+286)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = Float64(Float64(t / z) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((((t - x) / (a - z)) * (y - z)) + x) <= 1e+286)
		tmp = (t - x) + x;
	else
		tmp = (t / z) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 1e+286], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / z), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1.00000000000000003e286

   1. Initial program 78.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 24.1

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

   5. Applied rewrites 24.1%

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

   if 1.00000000000000003e286 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 82.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 56.0%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 27.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 20.4%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 10^{+286}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - a}{z}, x - t, t\right)\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y a) z) (- x t) t)))
   (if (<= z -7.5e-29)
     t_1
     (if (<= z 1.2e-41) (fma (/ (- y z) a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - a) / z), (x - t), t);
	double tmp;
	if (z <= -7.5e-29) {
		tmp = t_1;
	} else if (z <= 1.2e-41) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - a) / z), Float64(x - t), t)
	tmp = 0.0
	if (z <= -7.5e-29)
		tmp = t_1;
	elseif (z <= 1.2e-41)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.5e-29], t$95$1, If[LessEqual[z, 1.2e-41], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000006e-29 or 1.20000000000000011e-41 < z

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.7%

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

   if -7.50000000000000006e-29 < z < 1.20000000000000011e-41

   1. Initial program 90.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 82.1

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

   7. Applied rewrites 82.1%

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

Alternative 5: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - a}{z}, x - t, t\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y a) z) (- x t) t)))
   (if (<= z -7.2e-29) t_1 (if (<= z 1.2e-41) (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(((y - a) / z), (x - t), t);
	double tmp;
	if (z <= -7.2e-29) {
		tmp = t_1;
	} else if (z <= 1.2e-41) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - a) / z), Float64(x - t), t)
	tmp = 0.0
	if (z <= -7.2e-29)
		tmp = t_1;
	elseif (z <= 1.2e-41)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.2e-29], t$95$1, If[LessEqual[z, 1.2e-41], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.19999999999999948e-29 or 1.20000000000000011e-41 < z

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.7%

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

   if -7.19999999999999948e-29 < z < 1.20000000000000011e-41

   1. Initial program 90.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   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. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

Alternative 6: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) z) (- y a) t)))
   (if (<= z -8.8e-29) t_1 (if (<= z 1.25e-41) (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(((x - t) / z), (y - a), t);
	double tmp;
	if (z <= -8.8e-29) {
		tmp = t_1;
	} else if (z <= 1.25e-41) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -8.8e-29)
		tmp = t_1;
	elseif (z <= 1.25e-41)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -8.8e-29], t$95$1, If[LessEqual[z, 1.25e-41], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.79999999999999961e-29 or 1.2499999999999999e-41 < z

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 75.0%

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

   if -8.79999999999999961e-29 < z < 1.2499999999999999e-41

   1. Initial program 90.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

   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. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

Alternative 7: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -2.55 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -2.55e-42) t_1 (if (<= a 2.45e-32) (fma (/ y z) (- x t) t) 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 <= -2.55e-42) {
		tmp = t_1;
	} else if (a <= 2.45e-32) {
		tmp = fma((y / z), (x - t), t);
	} 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 <= -2.55e-42)
		tmp = t_1;
	elseif (a <= 2.45e-32)
		tmp = fma(Float64(y / z), Float64(x - t), t);
	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, -2.55e-42], t$95$1, If[LessEqual[a, 2.45e-32], N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.55e-42 or 2.4499999999999999e-32 < a

   1. Initial program 86.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -2.55e-42 < a < 2.4499999999999999e-32

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 78.7%

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

Alternative 8: 70.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.35:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.35)
   (fma (/ y z) (- x t) t)
   (if (<= z 5.5e-18) (fma (/ y a) (- t x) x) (fma (/ x z) (- y a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.35) {
		tmp = fma((y / z), (x - t), t);
	} else if (z <= 5.5e-18) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma((x / z), (y - a), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.35)
		tmp = fma(Float64(y / z), Float64(x - t), t);
	elseif (z <= 5.5e-18)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(x / z), Float64(y - a), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.35], N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 5.5e-18], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.34999999999999998

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 66.1%

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

   if -0.34999999999999998 < z < 5.5e-18

   1. Initial program 88.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if 5.5e-18 < z

   1. Initial program 64.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.4%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 73.4%

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

Alternative 9: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.35:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.35)
   (fma (/ y z) (- x t) t)
   (if (<= z 5.5e-18) (fma (/ (- t x) a) y x) (fma (/ x z) (- y a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.35) {
		tmp = fma((y / z), (x - t), t);
	} else if (z <= 5.5e-18) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = fma((x / z), (y - a), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.35)
		tmp = fma(Float64(y / z), Float64(x - t), t);
	elseif (z <= 5.5e-18)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = fma(Float64(x / z), Float64(y - a), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.35], N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 5.5e-18], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.34999999999999998

   1. Initial program 74.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.4%

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 66.1%

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

   if -0.34999999999999998 < z < 5.5e-18

   1. Initial program 88.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 5.5e-18 < z

   1. Initial program 64.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.4%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 73.4%

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

Alternative 10: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.1e-53)
   (fma (/ y z) (- x t) t)
   (if (<= z 1.02e-44) (+ (/ (* t y) a) x) (fma (/ x z) (- y a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e-53) {
		tmp = fma((y / z), (x - t), t);
	} else if (z <= 1.02e-44) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = fma((x / z), (y - a), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.1e-53)
		tmp = fma(Float64(y / z), Float64(x - t), t);
	elseif (z <= 1.02e-44)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = fma(Float64(x / z), Float64(y - a), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.1e-53], N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.02e-44], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.10000000000000009e-53

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 62.4%

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

   if -1.10000000000000009e-53 < z < 1.0199999999999999e-44

   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

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.3%

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

   if 1.0199999999999999e-44 < z

   1. Initial program 66.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 81.7%

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

   9. Applied rewrites 81.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 73.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x - t, t\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) (- y a) t)))
   (if (<= z -8.8e-29) t_1 (if (<= z 1.02e-44) (+ (/ (* t y) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / z), (y - a), t);
	double tmp;
	if (z <= -8.8e-29) {
		tmp = t_1;
	} else if (z <= 1.02e-44) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -8.8e-29)
		tmp = t_1;
	elseif (z <= 1.02e-44)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -8.8e-29], t$95$1, If[LessEqual[z, 1.02e-44], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.79999999999999961e-29 or 1.0199999999999999e-44 < z

   1. Initial program 69.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 75.0%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 67.5%

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

   if -8.79999999999999961e-29 < z < 1.0199999999999999e-44

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 64.4%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{if}\;z \leq -0.116:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) z) y t)))
   (if (<= z -0.116) t_1 (if (<= z 1.85e-91) (+ (/ (* t y) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / z), y, t);
	double tmp;
	if (z <= -0.116) {
		tmp = t_1;
	} else if (z <= 1.85e-91) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / z), y, t)
	tmp = 0.0
	if (z <= -0.116)
		tmp = t_1;
	elseif (z <= 1.85e-91)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[z, -0.116], t$95$1, If[LessEqual[z, 1.85e-91], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.116000000000000006 or 1.8500000000000001e-91 < z

   1. Initial program 72.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 65.1%

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

   if -0.116000000000000006 < z < 1.8500000000000001e-91

   1. Initial program 87.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.116:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -1.18 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -1.18e+110) t_1 (if (<= z 2.9e+66) (+ (/ (* t y) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -1.18e+110) {
		tmp = t_1;
	} else if (z <= 2.9e+66) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - x) + x
    if (z <= (-1.18d+110)) then
        tmp = t_1
    else if (z <= 2.9d+66) then
        tmp = ((t * y) / a) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -1.18e+110) {
		tmp = t_1;
	} else if (z <= 2.9e+66) {
		tmp = ((t * y) / a) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t - x) + x
	tmp = 0
	if z <= -1.18e+110:
		tmp = t_1
	elif z <= 2.9e+66:
		tmp = ((t * y) / a) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -1.18e+110)
		tmp = t_1;
	elseif (z <= 2.9e+66)
		tmp = Float64(Float64(Float64(t * y) / a) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (t - x) + x;
	tmp = 0.0;
	if (z <= -1.18e+110)
		tmp = t_1;
	elseif (z <= 2.9e+66)
		tmp = ((t * y) / a) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.18e+110], t$95$1, If[LessEqual[z, 2.9e+66], N[(N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1799999999999999e110 or 2.89999999999999986e66 < z

   1. Initial program 64.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 43.7

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

   5. Applied rewrites 43.7%

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

   if -1.1799999999999999e110 < z < 2.89999999999999986e66

   1. Initial program 86.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 63.6

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

   5. Applied rewrites 63.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.2%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+110}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{t \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - t}{z} \cdot y\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- x t) z) y)))
   (if (<= y -1.3e+123) t_1 (if (<= y 1.95e-23) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x - t) / z) * y;
	double tmp;
	if (y <= -1.3e+123) {
		tmp = t_1;
	} else if (y <= 1.95e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x - t) / z) * y
    if (y <= (-1.3d+123)) then
        tmp = t_1
    else if (y <= 1.95d-23) then
        tmp = (t - x) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x - t) / z) * y;
	double tmp;
	if (y <= -1.3e+123) {
		tmp = t_1;
	} else if (y <= 1.95e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((x - t) / z) * y
	tmp = 0
	if y <= -1.3e+123:
		tmp = t_1
	elif y <= 1.95e-23:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x - t) / z) * y)
	tmp = 0.0
	if (y <= -1.3e+123)
		tmp = t_1;
	elseif (y <= 1.95e-23)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x - t) / z) * y;
	tmp = 0.0;
	if (y <= -1.3e+123)
		tmp = t_1;
	elseif (y <= 1.95e-23)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.3e+123], t$95$1, If[LessEqual[y, 1.95e-23], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999993e123 or 1.95e-23 < y

   1. Initial program 89.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.3%

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

   if -1.29999999999999993e123 < y < 1.95e-23

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+123}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{y}{-z} \cdot t\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5.2e+124)
   (* (/ y (- z)) t)
   (if (<= y 1.8e-23) (+ (- t x) x) (* (/ y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5.2e+124) {
		tmp = (y / -z) * t;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = (y / z) * 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 (y <= (-5.2d+124)) then
        tmp = (y / -z) * t
    else if (y <= 1.8d-23) then
        tmp = (t - x) + x
    else
        tmp = (y / z) * 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 (y <= -5.2e+124) {
		tmp = (y / -z) * t;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -5.2e+124:
		tmp = (y / -z) * t
	elif y <= 1.8e-23:
		tmp = (t - x) + x
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -5.2e+124)
		tmp = Float64(Float64(y / Float64(-z)) * t);
	elseif (y <= 1.8e-23)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -5.2e+124)
		tmp = (y / -z) * t;
	elseif (y <= 1.8e-23)
		tmp = (t - x) + x;
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -5.2e+124], N[(N[(y / (-z)), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[y, 1.8e-23], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2000000000000001e124

   1. Initial program 94.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 60.2%

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 56.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 36.7%

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

   if -5.2000000000000001e124 < y < 1.7999999999999999e-23

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 31.7

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

   5. Applied rewrites 31.7%

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

   if 1.7999999999999999e-23 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 27.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{y}{-z} \cdot t\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.8e+131)
   (* (/ x z) y)
   (if (<= y 1.8e-23) (+ (- t x) x) (* (/ y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.8e+131) {
		tmp = (x / z) * y;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = (y / z) * 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 (y <= (-2.8d+131)) then
        tmp = (x / z) * y
    else if (y <= 1.8d-23) then
        tmp = (t - x) + x
    else
        tmp = (y / z) * 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 (y <= -2.8e+131) {
		tmp = (x / z) * y;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.8e+131:
		tmp = (x / z) * y
	elif y <= 1.8e-23:
		tmp = (t - x) + x
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.8e+131)
		tmp = Float64(Float64(x / z) * y);
	elseif (y <= 1.8e-23)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2.8e+131)
		tmp = (x / z) * y;
	elseif (y <= 1.8e-23)
		tmp = (t - x) + x;
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.8e+131], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.8e-23], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8000000000000001e131

   1. Initial program 93.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 54.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 33.3%

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

   if -2.8000000000000001e131 < y < 1.7999999999999999e-23

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 31.5

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

   5. Applied rewrites 31.5%

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

   if 1.7999999999999999e-23 < y

   1. Initial program 87.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 37.7%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 27.0%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 27.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot x\\ \mathbf{if}\;y \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y z) x)))
   (if (<= y -2.8e+131) t_1 (if (<= y 1.8e-23) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / z) * x;
	double tmp;
	if (y <= -2.8e+131) {
		tmp = t_1;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y / z) * x
    if (y <= (-2.8d+131)) then
        tmp = t_1
    else if (y <= 1.8d-23) then
        tmp = (t - x) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / z) * x;
	double tmp;
	if (y <= -2.8e+131) {
		tmp = t_1;
	} else if (y <= 1.8e-23) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / z) * x
	tmp = 0
	if y <= -2.8e+131:
		tmp = t_1
	elif y <= 1.8e-23:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / z) * x)
	tmp = 0.0
	if (y <= -2.8e+131)
		tmp = t_1;
	elseif (y <= 1.8e-23)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / z) * x;
	tmp = 0.0;
	if (y <= -2.8e+131)
		tmp = t_1;
	elseif (y <= 1.8e-23)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -2.8e+131], t$95$1, If[LessEqual[y, 1.8e-23], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8000000000000001e131 or 1.7999999999999999e-23 < y

   1. Initial program 89.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 43.8%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 29.2%

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

   if -2.8000000000000001e131 < y < 1.7999999999999999e-23

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 31.5

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

   5. Applied rewrites 31.5%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(t - x\right) + x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 21.7

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

5. Applied rewrites 21.7%

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

6. Final simplification 21.7%

   \[\leadsto \left(t - x\right) + x \]
7. Add Preprocessing

Alternative 19: 2.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) + x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 21.7

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

5. Applied rewrites 21.7%

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

6. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
7. Step-by-step derivation

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]

9. Final simplification 2.8%

   \[\leadsto \left(-x\right) + x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024295 
(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)))))