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

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 10

Speedup: 1.0×

• 20.8% 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: 99.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (((x - y) - (0.5 * z)) * 4.0) / 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 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $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. Final simplification 100.0%

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

Alternative 2: 66.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ t_1 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;t\_0 \leq -1000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)) (t_1 (/ (* -4.0 y) z)))
   (if (<= t_0 -5e+55)
     (/ (* x 4.0) z)
     (if (<= t_0 -1000000.0) t_1 (if (<= t_0 -1.0) -2.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_1 = (-4.0 * y) / z;
	double tmp;
	if (t_0 <= -5e+55) {
		tmp = (x * 4.0) / z;
	} else if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_1;
	}
	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 - y) - (0.5d0 * z)) * 4.0d0) / z
    t_1 = ((-4.0d0) * y) / z
    if (t_0 <= (-5d+55)) then
        tmp = (x * 4.0d0) / z
    else if (t_0 <= (-1000000.0d0)) then
        tmp = t_1
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_1 = (-4.0 * y) / z;
	double tmp;
	if (t_0 <= -5e+55) {
		tmp = (x * 4.0) / z;
	} else if (t_0 <= -1000000.0) {
		tmp = t_1;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - y) - (0.5 * z)) * 4.0) / z
	t_1 = (-4.0 * y) / z
	tmp = 0
	if t_0 <= -5e+55:
		tmp = (x * 4.0) / z
	elif t_0 <= -1000000.0:
		tmp = t_1
	elif t_0 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	t_1 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (t_0 <= -5e+55)
		tmp = Float64(Float64(x * 4.0) / z);
	elseif (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - y) - (0.5 * z)) * 4.0) / z;
	t_1 = (-4.0 * y) / z;
	tmp = 0.0;
	if (t_0 <= -5e+55)
		tmp = (x * 4.0) / z;
	elseif (t_0 <= -1000000.0)
		tmp = t_1;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+55], N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, -1000000.0], t$95$1, If[LessEqual[t$95$0, -1.0], -2.0, t$95$1]]]]]

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) < -5.00000000000000046e55

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 59.9

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

   5. Applied rewrites 59.9%

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

   if -5.00000000000000046e55 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e6 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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

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

      1. lower-*.f64 56.2

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

   5. Applied rewrites 56.2%

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

   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. Applied rewrites 97.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1000000:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) 4.0) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -1000000.0)
     t_0
     (if (<= t_1 1000000.0) (fma (/ 4.0 z) x -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - y) * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1000000.0) {
		tmp = fma((4.0 / z), x, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - y) * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 1000000.0)
		tmp = fma(Float64(4.0 / z), x, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 1000000.0], N[(N[(4.0 / z), $MachinePrecision] * x + -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) < -1e6 or 1e6 < (/.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

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

      1. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   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. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.6%

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

Alternative 4: 98.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \frac{4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x y) (/ 4.0 z))) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -1000000.0)
     t_0
     (if (<= t_1 1000000.0) (fma (/ 4.0 z) x -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) * (4.0 / z);
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_0;
	} else if (t_1 <= 1000000.0) {
		tmp = fma((4.0 / z), x, -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(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_0;
	elseif (t_1 <= 1000000.0)
		tmp = fma(Float64(4.0 / z), x, -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[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$0, If[LessEqual[t$95$1, 1000000.0], N[(N[(4.0 / z), $MachinePrecision] * x + -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) < -1e6 or 1e6 < (/.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

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

      1. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. times-frac N/A

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

      5. /-rgt-identity N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.2

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

   7. Applied rewrites 99.2%

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

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

   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. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.5%

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

Alternative 5: 66.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 y) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -1000000.0) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -1000000.0) {
		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 = ((-4.0d0) * y) / z
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-1000000.0d0)) 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 = (-4.0 * y) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -1000000.0) {
		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 = (-4.0 * y) / z
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -1000000.0:
		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(Float64(-4.0 * y) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		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 = (-4.0 * y) / z;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		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[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], 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) < -1e6 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 y around inf

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

      1. lower-*.f64 53.5

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

   5. Applied rewrites 53.5%

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

   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. Applied rewrites 97.9%

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

4. Final simplification 69.8%

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

Alternative 6: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{if}\;y \leq -8.4 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ y z) -4.0 -2.0)))
   (if (<= y -8.4e+85) t_0 (if (<= y 4.6e+61) (fma (/ x z) 4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y / z), -4.0, -2.0);
	double tmp;
	if (y <= -8.4e+85) {
		tmp = t_0;
	} else if (y <= 4.6e+61) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y / z), -4.0, -2.0)
	tmp = 0.0
	if (y <= -8.4e+85)
		tmp = t_0;
	elseif (y <= 4.6e+61)
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision]}, If[LessEqual[y, -8.4e+85], t$95$0, If[LessEqual[y, 4.6e+61], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4000000000000004e85 or 4.5999999999999999e61 < 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 z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

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

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt-in N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 90.6

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

   8. Applied rewrites 90.6%

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

   if -8.4000000000000004e85 < y < 4.5999999999999999e61

   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 z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   7. 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. associate-/l* N/A

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

      3. *-inverses N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 91.9

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

   8. Applied rewrites 91.9%

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

Alternative 7: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 y) z)))
   (if (<= y -2.25e+86) t_0 (if (<= y 2.3e+95) (fma (/ x z) 4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double tmp;
	if (y <= -2.25e+86) {
		tmp = t_0;
	} else if (y <= 2.3e+95) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (y <= -2.25e+86)
		tmp = t_0;
	elseif (y <= 2.3e+95)
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -2.25e+86], t$95$0, If[LessEqual[y, 2.3e+95], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.24999999999999996e86 or 2.29999999999999997e95 < 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

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

      1. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -2.24999999999999996e86 < y < 2.29999999999999997e95

   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 z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   7. 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. associate-/l* N/A

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

      3. *-inverses N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 92.1

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

   8. Applied rewrites 92.1%

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

Alternative 8: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 y) z)))
   (if (<= y -2.25e+86) t_0 (if (<= y 2.3e+95) (fma (/ 4.0 z) x -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * y) / z;
	double tmp;
	if (y <= -2.25e+86) {
		tmp = t_0;
	} else if (y <= 2.3e+95) {
		tmp = fma((4.0 / z), x, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (y <= -2.25e+86)
		tmp = t_0;
	elseif (y <= 2.3e+95)
		tmp = fma(Float64(4.0 / z), x, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -2.25e+86], t$95$0, If[LessEqual[y, 2.3e+95], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.24999999999999996e86 or 2.29999999999999997e95 < 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

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

      1. lower-*.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -2.24999999999999996e86 < y < 2.29999999999999997e95

   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. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 92.0

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

   5. Applied rewrites 92.0%

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

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. associate-*l/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-fma.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 10: 33.5% 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. Applied rewrites 37.7%

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

Developer Target 1: 97.9% 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 2024268 
(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))