Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 10

Speedup: 1.0×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - y\right)}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -40000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 20000000000000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x y)) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -40000000000000.0)
     t_0
     (if (<= t_1 20000000000000.0) (fma 4.0 (/ x z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - y)) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -40000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 20000000000000.0) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - y)) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -40000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 20000000000000.0)
		tmp = fma(4.0, Float64(x / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -40000000000000.0], t$95$0, If[LessEqual[t$95$1, 20000000000000.0], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4e13 or 2e13 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)} \cdot \frac{x - y}{z} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(-4 \cdot \frac{x - y}{z}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{x - y}{z} \cdot -4}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\left(x - y\right) \cdot -4}{z}}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{-4 \cdot \left(x - y\right)}}{z}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(-4 \cdot \left(x - y\right)\right) \cdot 1}}{z}\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(-4 \cdot \left(x - y\right)\right) \cdot \color{blue}{\frac{z}{z}}}{z}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\left(-4 \cdot \left(x - y\right)\right) \cdot z}{z}}}{z}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{-4 \cdot \left(x - y\right)}{z} \cdot z}}{z}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(-4 \cdot \frac{x - y}{z}\right)} \cdot z}{z}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{z \cdot \left(-4 \cdot \frac{x - y}{z}\right)}}{z}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z \cdot \left(-4 \cdot \frac{x - y}{z}\right)\right)}{z}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z \cdot \left(-4 \cdot \frac{x - y}{z}\right)\right)}{z}} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z}} \]

   if -4e13 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e13

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{-2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

      10. lower-/.f64 97.2

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z}}, -2\right) \]

   5. Simplified 97.2%

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

Alternative 3: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -40000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 20000000000000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x y) (/ 4.0 z))) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -40000000000000.0)
     t_0
     (if (<= t_1 20000000000000.0) (fma 4.0 (/ x z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) * (4.0 / z);
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -40000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 20000000000000.0) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) * Float64(4.0 / z))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -40000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 20000000000000.0)
		tmp = fma(4.0, Float64(x / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -40000000000000.0], t$95$0, If[LessEqual[t$95$1, 20000000000000.0], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -4e13 or 2e13 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-4\right)\right)} \cdot \frac{x - y}{z} \]

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

         \[\leadsto \color{blue}{\mathsf{neg}\left(-4 \cdot \frac{x - y}{z}\right)} \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{x - y}{z} \cdot -4}\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\left(x - y\right) \cdot -4}{z}}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{-4 \cdot \left(x - y\right)}}{z}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(-4 \cdot \left(x - y\right)\right) \cdot 1}}{z}\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left(-4 \cdot \left(x - y\right)\right) \cdot \color{blue}{\frac{z}{z}}}{z}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{\left(-4 \cdot \left(x - y\right)\right) \cdot z}{z}}}{z}\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\frac{-4 \cdot \left(x - y\right)}{z} \cdot z}}{z}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{\left(-4 \cdot \frac{x - y}{z}\right)} \cdot z}{z}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{z \cdot \left(-4 \cdot \frac{x - y}{z}\right)}}{z}\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z \cdot \left(-4 \cdot \frac{x - y}{z}\right)\right)}{z}} \]

      13. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z \cdot \left(-4 \cdot \frac{x - y}{z}\right)\right)}{z}} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(x - y\right)}{z}} \]

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - y\right)} \]

   if -4e13 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e13

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{-2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

      10. lower-/.f64 97.2

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z}}, -2\right) \]

   5. Simplified 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -40000000000000:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 20000000000000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_0 -1000000.0)
     (/ (* y -4.0) z)
     (if (<= t_0 -1.0) -2.0 (/ (* 4.0 x) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = (4.0 * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_0 <= (-1000000.0d0)) then
        tmp = (y * (-4.0d0)) / z
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = (4.0d0 * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = (4.0 * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_0 <= -1000000.0:
		tmp = (y * -4.0) / z
	elif t_0 <= -1.0:
		tmp = -2.0
	else:
		tmp = (4.0 * x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(Float64(y * -4.0) / z);
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = Float64(Float64(4.0 * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_0 <= -1000000.0)
		tmp = (y * -4.0) / z;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = (4.0 * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, -1.0], -2.0, N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. lower-*.f64 57.2

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

   if -1e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 95.7%

      \[\leadsto \color{blue}{-2} \]

   if -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      3. lower-*.f64 55.9

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Simplified 55.9%

      \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1000000:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -1000000:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_0 -1000000.0)
     (/ (* y -4.0) z)
     (if (<= t_0 -1.0) -2.0 (* x (/ 4.0 z))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = x * (4.0 / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_0 <= (-1000000.0d0)) then
        tmp = (y * (-4.0d0)) / z
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = x * (4.0d0 / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (y * -4.0) / z;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = x * (4.0 / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_0 <= -1000000.0:
		tmp = (y * -4.0) / z
	elif t_0 <= -1.0:
		tmp = -2.0
	else:
		tmp = x * (4.0 / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(Float64(y * -4.0) / z);
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = Float64(x * Float64(4.0 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_0 <= -1000000.0)
		tmp = (y * -4.0) / z;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = x * (4.0 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, -1.0], -2.0, N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. lower-*.f64 57.2

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

   if -1e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 95.7%

      \[\leadsto \color{blue}{-2} \]

   if -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      3. lower-*.f64 55.9

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Simplified 55.9%

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

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4}{z}} \cdot x \]

      3. lower-*.f64 55.8

         \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

   7. Applied egg-rr 55.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1000000:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -5000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 z))) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -5000000.0) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / z);
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (4.0d0 / z)
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-5000000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (4.0 / z);
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -5000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / z)
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -5000000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / z))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -5000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (4.0 / z);
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -5000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5000000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5e6 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4 \cdot x}{z}} \]

      3. lower-*.f64 51.7

         \[\leadsto \frac{\color{blue}{4 \cdot x}}{z} \]

   5. Simplified 51.7%

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

      1. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{4}{z}} \cdot x \]

      3. lower-*.f64 51.6

         \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

   7. Applied egg-rr 51.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot x} \]

   if -5e6 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Simplified 94.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5000000:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ t_1 := \mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y -4.0) z)) (t_1 (fma 4.0 (/ x z) -2.0)))
   (if (<= y -1.12e+174)
     t_0
     (if (<= y -5e+112)
       t_1
       (if (<= y -4.8e+49) t_0 (if (<= y 4.4e+122) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * -4.0) / z;
	double t_1 = fma(4.0, (x / z), -2.0);
	double tmp;
	if (y <= -1.12e+174) {
		tmp = t_0;
	} else if (y <= -5e+112) {
		tmp = t_1;
	} else if (y <= -4.8e+49) {
		tmp = t_0;
	} else if (y <= 4.4e+122) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * -4.0) / z)
	t_1 = fma(4.0, Float64(x / z), -2.0)
	tmp = 0.0
	if (y <= -1.12e+174)
		tmp = t_0;
	elseif (y <= -5e+112)
		tmp = t_1;
	elseif (y <= -4.8e+49)
		tmp = t_0;
	elseif (y <= 4.4e+122)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[y, -1.12e+174], t$95$0, If[LessEqual[y, -5e+112], t$95$1, If[LessEqual[y, -4.8e+49], t$95$0, If[LessEqual[y, 4.4e+122], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.11999999999999993e174 or -5e112 < y < -4.8e49 or 4.3999999999999998e122 < y

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

      3. lower-*.f64 77.6

         \[\leadsto \frac{\color{blue}{-4 \cdot y}}{z} \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} \]

   if -1.11999999999999993e174 < y < -5e112 or -4.8e49 < y < 4.3999999999999998e122

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{-2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

      10. lower-/.f64 89.7

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z}}, -2\right) \]

   5. Simplified 89.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+174}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \frac{-4}{z}, -2\right)\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (/ -4.0 z) -2.0)))
   (if (<= y -1.55e+48) t_0 (if (<= y 3.7e+34) (fma 4.0 (/ x z) -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (-4.0 / z), -2.0);
	double tmp;
	if (y <= -1.55e+48) {
		tmp = t_0;
	} else if (y <= 3.7e+34) {
		tmp = fma(4.0, (x / z), -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(-4.0 / z), -2.0)
	tmp = 0.0
	if (y <= -1.55e+48)
		tmp = t_0;
	elseif (y <= 3.7e+34)
		tmp = fma(4.0, Float64(x / z), -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(-4.0 / z), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[y, -1.55e+48], t$95$0, If[LessEqual[y, 3.7e+34], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000003e48 or 3.70000000000000009e34 < y

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x - y, -2\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} - 2} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{-4 \cdot \frac{y}{z} + \left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2} \]

      3. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{-2 \cdot 1} \]

      4. lft-mult-inverse N/A

         \[\leadsto -4 \cdot \frac{y}{z} + -2 \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)} \]

      5. associate-*l* N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(-2 \cdot \frac{1}{x}\right) \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto -4 \cdot \frac{y}{z} + \left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \cdot \frac{1}{x}\right) \cdot x \]

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

         \[\leadsto -4 \cdot \frac{y}{z} + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right)} \cdot x \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot y}{z}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot -4}}{z} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      10. associate-/l* N/A

          \[\leadsto \color{blue}{y \cdot \frac{-4}{z}} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      11. metadata-eval N/A

          \[\leadsto y \cdot \frac{\color{blue}{\mathsf{neg}\left(4\right)}}{z} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      12. distribute-neg-frac N/A

          \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{4}{z}\right)\right)} + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      13. metadata-eval N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{4 \cdot 1}}{z}\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      14. associate-*r/ N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{4 \cdot \frac{1}{z}}\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(4 \cdot \frac{1}{z}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right)} \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{neg}\left(\color{blue}{\frac{4 \cdot 1}{z}}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{neg}\left(\frac{\color{blue}{4}}{z}\right), \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      18. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\mathsf{neg}\left(4\right)}{z}}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{-4}}{z}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

      20. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-4}{z}}, \left(\mathsf{neg}\left(2 \cdot \frac{1}{x}\right)\right) \cdot x\right) \]

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

          \[\leadsto \mathsf{fma}\left(y, \frac{-4}{z}, \color{blue}{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{x}\right)} \cdot x\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-4}{z}, \left(\color{blue}{-2} \cdot \frac{1}{x}\right) \cdot x\right) \]

      23. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{-4}{z}, \color{blue}{-2 \cdot \left(\frac{1}{x} \cdot x\right)}\right) \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-4}{z}, -2\right)} \]

   if -1.55000000000000003e48 < y < 3.70000000000000009e34

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} - \frac{\frac{1}{2} \cdot z}{z}\right)} \]

      2. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right)} \]

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \frac{z}{z}}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \color{blue}{1}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + 4 \cdot \color{blue}{\frac{-1}{2}} \]

      8. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{-2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{z}, -2\right)} \]

      10. lower-/.f64 90.4

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{x}{z}}, -2\right) \]

   5. Simplified 90.4%

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

Alternative 9: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{4}{z}, x - y, -2\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ 4.0 z) (- x y) -2.0))

C

double code(double x, double y, double z) {
	return fma((4.0 / z), (x - y), -2.0);
}

Julia

function code(x, y, z)
	return fma(Float64(4.0 / z), Float64(x - y), -2.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 / z), $MachinePrecision] * N[(x - y), $MachinePrecision] + -2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z} + 4 \cdot \frac{x}{z}} \]

5. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x - y, -2\right)} \]
6. Add Preprocessing

Alternative 10: 33.6% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 100.0%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Simplified 37.2%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer Target 1: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))