Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 96.8% → 96.8%

Time: 10.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-inv N/A

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

   4. lift--.f64 N/A

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

   5. flip3-- N/A

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

   6. clear-num N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. clear-num N/A

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

   12. flip3-- N/A

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

   13. lift--.f64 N/A

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

   14. associate-*r* N/A

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

   15. div-inv N/A

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

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

   \[\leadsto \mathsf{fma}\left(\frac{x}{z - y}, t, t \cdot \frac{y}{y - z}\right) \]
6. Add Preprocessing

Alternative 2: 81.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z - y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;-t \cdot \frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t (- z y)))))
   (if (<= t_1 -1e-61)
     t_2
     (if (<= t_1 -2e-138)
       (- (* t (/ y z)))
       (if (<= t_1 0.95)
         (* t (/ x z))
         (if (<= t_1 2.5) (fma t (/ z y) t) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / (z - y));
	double tmp;
	if (t_1 <= -1e-61) {
		tmp = t_2;
	} else if (t_1 <= -2e-138) {
		tmp = -(t * (y / z));
	} else if (t_1 <= 0.95) {
		tmp = t * (x / z);
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x * Float64(t / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -1e-61)
		tmp = t_2;
	elseif (t_1 <= -2e-138)
		tmp = Float64(-Float64(t * Float64(y / z)));
	elseif (t_1 <= 0.95)
		tmp = Float64(t * Float64(x / z));
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-61], t$95$2, If[LessEqual[t$95$1, -2e-138], (-N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), If[LessEqual[t$95$1, 0.95], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-61 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.7

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

   5. Simplified 93.7%

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

      1. lift--.f64 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 86.7

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

   7. Applied egg-rr 86.7%

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

   if -1e-61 < (/.f64 (-.f64 x y) (-.f64 z y)) < -2.00000000000000013e-138

   1. Initial program 99.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.0

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

   8. Simplified 77.0%

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

   if -2.00000000000000013e-138 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 92.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 68.9

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

   5. Simplified 68.9%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.2

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

   8. Simplified 98.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;-t \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 0.95)
       (* t (/ (- x y) z))
       (if (<= t_1 2.5) (fma t (/ (- z x) y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.95) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.5) {
		tmp = fma(t, ((z - x) / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.95)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(Float64(z - x) / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, 0.95], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5], N[(t * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.3

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

   5. Simplified 96.3%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 94.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.7

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

   5. Simplified 93.7%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 0.95)
       (* t (/ (- x y) z))
       (if (<= t_1 2.5) (fma t (/ x (- y)) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.95) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (x / -y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.95)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(x / Float64(-y)), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, 0.95], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5], N[(t * N[(x / (-y)), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.3

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

   5. Simplified 96.3%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 94.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.7

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

   5. Simplified 93.7%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-in N/A

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

      8. neg-mul-1 N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-mul-1 N/A

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

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

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

      14. *-commutative N/A

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

      15. neg-mul-1 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. mul-1-neg N/A

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

      19. distribute-neg-frac2 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 99.0

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

   5. Simplified 99.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{-y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 0.95)
       (* t (/ (- x y) z))
       (if (<= t_1 2.0) (fma t (/ z y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.95) {
		tmp = t * ((x - y) / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.95)
		tmp = Float64(t * Float64(Float64(x - y) / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, 0.95], N[(t * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.4

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

   5. Simplified 95.4%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 94.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.7

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

   5. Simplified 93.7%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 82.4

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

   5. Simplified 82.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.2

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

   8. Simplified 99.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y} \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* (/ x (- z y)) t)))
   (if (<= t_1 -2e-6)
     t_2
     (if (<= t_1 0.95)
       (* (- x y) (/ t z))
       (if (<= t_1 2.0) (fma t (/ z y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = (x / (z - y)) * t;
	double tmp;
	if (t_1 <= -2e-6) {
		tmp = t_2;
	} else if (t_1 <= 0.95) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.0) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(Float64(x / Float64(z - y)) * t)
	tmp = 0.0
	if (t_1 <= -2e-6)
		tmp = t_2;
	elseif (t_1 <= 0.95)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.0)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-6], t$95$2, If[LessEqual[t$95$1, 0.95], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.99999999999999991e-6 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.5

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

   5. Simplified 95.5%

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

   if -1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 87.3

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

   5. Simplified 87.3%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 82.4

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

   5. Simplified 82.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.2

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

   8. Simplified 99.2%

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

Alternative 7: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z - y}\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.95:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t (- z y)))))
   (if (<= t_1 -5000000000.0)
     t_2
     (if (<= t_1 0.95)
       (* (- x y) (/ t z))
       (if (<= t_1 2.5) (fma t (/ z y) t) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / (z - y));
	double tmp;
	if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.95) {
		tmp = (x - y) * (t / z);
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x * Float64(t / Float64(z - y)))
	tmp = 0.0
	if (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= 0.95)
		tmp = Float64(Float64(x - y) * Float64(t / z));
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, 0.95], N[(N[(x - y), $MachinePrecision] * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.5], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 98.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 96.3

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

   5. Simplified 96.3%

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

      1. lift--.f64 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.6

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

   7. Applied egg-rr 90.6%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996

   1. Initial program 94.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 85.6

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

   5. Simplified 85.6%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.2

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

   8. Simplified 98.2%

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

Alternative 8: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq 0.95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 0.95)
     t_2
     (if (<= t_1 2.5)
       (fma t (/ z y) t)
       (if (<= t_1 4e+30) t_2 (* t (/ x (- y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 0.95) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (z / y), t);
	} else if (t_1 <= 4e+30) {
		tmp = t_2;
	} else {
		tmp = t * (x / -y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (t_1 <= 0.95)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(z / y), t);
	elseif (t_1 <= 4e+30)
		tmp = t_2;
	else
		tmp = Float64(t * Float64(x / Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.95], t$95$2, If[LessEqual[t$95$1, 2.5], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 4e+30], t$95$2, N[(t * N[(x / (-y)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.0000000000000001e30

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.2

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

   5. Simplified 61.2%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.2

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

   8. Simplified 98.2%

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

   if 4.0000000000000001e30 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 70.3

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

   8. Simplified 70.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq 0.95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+30}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{-y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 0.95)
     t_2
     (if (<= t_1 2.5)
       (fma t (/ z y) t)
       (if (<= t_1 4e+30) t_2 (* x (/ t (- y))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 0.95) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (z / y), t);
	} else if (t_1 <= 4e+30) {
		tmp = t_2;
	} else {
		tmp = x * (t / -y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (t_1 <= 0.95)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(z / y), t);
	elseif (t_1 <= 4e+30)
		tmp = t_2;
	else
		tmp = Float64(x * Float64(t / Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.95], t$95$2, If[LessEqual[t$95$1, 2.5], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 4e+30], t$95$2, N[(x * N[(t / (-y)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.0000000000000001e30

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.2

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

   5. Simplified 61.2%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.2

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

   8. Simplified 98.2%

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

   if 4.0000000000000001e30 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Simplified 99.8%

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

      1. lift--.f64 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 95.9

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

   7. Applied egg-rr 95.9%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.1

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

   10. Simplified 70.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq 0.95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 0.95) t_2 (if (<= t_1 2.5) (fma t (/ z y) t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 0.95) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = fma(t, (z / y), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (t_1 <= 0.95)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = fma(t, Float64(z / y), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.95], t$95$2, If[LessEqual[t$95$1, 2.5], N[(t * N[(z / y), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.9

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

   5. Simplified 57.9%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lower-/.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. lower-neg.f64 81.4

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

   5. Simplified 81.4%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.2

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

   8. Simplified 98.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;t\_1 \leq 0.95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* t (/ x z))))
   (if (<= t_1 0.95) t_2 (if (<= t_1 2.5) t t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 0.95) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = t * (x / z)
    if (t_1 <= 0.95d0) then
        tmp = t_2
    else if (t_1 <= 2.5d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = t * (x / z);
	double tmp;
	if (t_1 <= 0.95) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = t * (x / z)
	tmp = 0
	if t_1 <= 0.95:
		tmp = t_2
	elif t_1 <= 2.5:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (t_1 <= 0.95)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = t * (x / z);
	tmp = 0.0;
	if (t_1 <= 0.95)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.95], t$95$2, If[LessEqual[t$95$1, 2.5], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.94999999999999996 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.9

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

   5. Simplified 57.9%

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

   if 0.94999999999999996 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Simplified 97.2%

      \[\leadsto \color{blue}{1} \cdot t \]
   6. Step-by-step derivation

      1. *-lft-identity 97.2

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.95:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2.5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y}{z - y}\\ t_2 := x \cdot \frac{t}{z}\\ \mathbf{if}\;t\_1 \leq 0.001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2.5:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x y) (- z y))) (t_2 (* x (/ t z))))
   (if (<= t_1 0.001) t_2 (if (<= t_1 2.5) t t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / z);
	double tmp;
	if (t_1 <= 0.001) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x - y) / (z - y)
    t_2 = x * (t / z)
    if (t_1 <= 0.001d0) then
        tmp = t_2
    else if (t_1 <= 2.5d0) then
        tmp = t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - y) / (z - y);
	double t_2 = x * (t / z);
	double tmp;
	if (t_1 <= 0.001) {
		tmp = t_2;
	} else if (t_1 <= 2.5) {
		tmp = t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - y) / (z - y)
	t_2 = x * (t / z)
	tmp = 0
	if t_1 <= 0.001:
		tmp = t_2
	elif t_1 <= 2.5:
		tmp = t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (t_1 <= 0.001)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = t;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - y) / (z - y);
	t_2 = x * (t / z);
	tmp = 0.0;
	if (t_1 <= 0.001)
		tmp = t_2;
	elseif (t_1 <= 2.5)
		tmp = t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.001], t$95$2, If[LessEqual[t$95$1, 2.5], t, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-3 or 2.5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 78.9

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

   5. Simplified 78.9%

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

      1. lift--.f64 N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 75.0

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

   7. Applied egg-rr 75.0%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 54.8

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

   10. Simplified 54.8%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.5

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Simplified 96.4%

      \[\leadsto \color{blue}{1} \cdot t \]
   6. Step-by-step derivation

      1. *-lft-identity 96.4

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

   7. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. div-sub N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 97.7

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

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

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

Alternative 14: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ t \cdot \frac{x - y}{z - y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Final simplification 97.7%

   \[\leadsto t \cdot \frac{x - y}{z - y} \]
4. Add Preprocessing

Alternative 15: 34.2% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 t)

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return t

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 97.7%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \cdot t \]
4. Step-by-step derivation

5. Simplified 34.7%

   \[\leadsto \color{blue}{1} \cdot t \]
6. Step-by-step derivation

   1. *-lft-identity 34.7

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

7. Applied egg-rr 34.7%

   \[\leadsto \color{blue}{t} \]
8. Add Preprocessing

Developer Target 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\frac{z - y}{x - y}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ t (/ (- z y) (- x y))))

  (* (/ (- x y) (- z y)) t))