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

Percentage Accurate: 99.8% → 100.0%

Time: 7.2s

Alternatives: 9

Speedup: 1.3×

• 20.2% 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: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * 4.0 + -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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

      \[\leadsto \frac{4}{\color{blue}{\frac{z}{\left(x - y\right) - z \cdot 0.5}}} \]

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 99.8

       \[\leadsto \frac{4}{\frac{z}{\mathsf{fma}\left(\color{blue}{-0.5}, z, x - y\right)}} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{4}{\frac{z}{\mathsf{fma}\left(-0.5, z, x - y\right)}}} \]

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z} - 2} \]
6. 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. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 66.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ t_2 := \frac{-4 \cdot y}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+184}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z))
        (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z))
        (t_2 (/ (* -4.0 y) z)))
   (if (<= t_1 -1e+222)
     t_0
     (if (<= t_1 -5000000000.0)
       t_2
       (if (<= t_1 -1.0)
         -2.0
         (if (<= t_1 2e+184) t_0 (if (<= t_1 1e+284) t_2 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -1e+222) {
		tmp = t_0;
	} else if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+184) {
		tmp = t_0;
	} else if (t_1 <= 1e+284) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (x * 4.0d0) / z
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    t_2 = ((-4.0d0) * y) / z
    if (t_1 <= (-1d+222)) then
        tmp = t_0
    else if (t_1 <= (-5000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 2d+184) then
        tmp = t_0
    else if (t_1 <= 1d+284) then
        tmp = t_2
    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 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double t_2 = (-4.0 * y) / z;
	double tmp;
	if (t_1 <= -1e+222) {
		tmp = t_0;
	} else if (t_1 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 2e+184) {
		tmp = t_0;
	} else if (t_1 <= 1e+284) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / z
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	t_2 = (-4.0 * y) / z
	tmp = 0
	if t_1 <= -1e+222:
		tmp = t_0
	elif t_1 <= -5000000000.0:
		tmp = t_2
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 2e+184:
		tmp = t_0
	elif t_1 <= 1e+284:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	t_2 = Float64(Float64(-4.0 * y) / z)
	tmp = 0.0
	if (t_1 <= -1e+222)
		tmp = t_0;
	elseif (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+184)
		tmp = t_0;
	elseif (t_1 <= 1e+284)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / z;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	t_2 = (-4.0 * y) / z;
	tmp = 0.0;
	if (t_1 <= -1e+222)
		tmp = t_0;
	elseif (t_1 <= -5000000000.0)
		tmp = t_2;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 2e+184)
		tmp = t_0;
	elseif (t_1 <= 1e+284)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 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]}, Block[{t$95$2 = N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+222], t$95$0, If[LessEqual[t$95$1, -5000000000.0], t$95$2, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 2e+184], t$95$0, If[LessEqual[t$95$1, 1e+284], t$95$2, t$95$0]]]]]]]]

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) < -1e222 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2.00000000000000003e184 or 1.00000000000000008e284 < (/.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

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

      1. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -1e222 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5e9 or 2.00000000000000003e184 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.00000000000000008e284

   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -5e9 < (/.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 93.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1 \cdot 10^{+222}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -5000000000:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 2 \cdot 10^{+184}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+284}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot \left(y - x\right)}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -20:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* -4.0 (- y x)) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -20.0) t_0 (if (<= t_1 1e+16) (fma (/ y z) -4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-4.0 * (y - x)) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -20.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+16) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-4.0 * Float64(y - x)) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -20.0)
		tmp = t_0;
	elseif (t_1 <= 1e+16)
		tmp = fma(Float64(y / 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 * N[(y - x), $MachinePrecision]), $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, -20.0], t$95$0, If[LessEqual[t$95$1, 1e+16], N[(N[(y / z), $MachinePrecision] * -4.0 + -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) < -20 or 1e16 < (/.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

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

      1. sub-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. neg-mul-1 N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-in N/A

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

      15. sub-neg N/A

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

      16. *-rgt-identity N/A

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

      17. *-inverses N/A

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

      18. associate-/l* N/A

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

      19. associate-*l/ N/A

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

      20. associate-*r/ N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. associate-*r* N/A

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

   5. Applied rewrites 99.0%

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

   if -20 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1e16

   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 5.2

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

   5. Applied rewrites 5.2%

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

   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. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. associate-*r/ N/A

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

      14. div-sub N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-in N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

   8. Applied rewrites 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -20:\\ \;\;\;\;\frac{-4 \cdot \left(y - x\right)}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(y - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-4 \cdot y}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -5000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -5000000000.0) 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 <= -5000000000.0) {
		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 <= (-5000000000.0d0)) 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 <= -5000000000.0) {
		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 <= -5000000000.0:
		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 <= -5000000000.0)
		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 <= -5000000000.0)
		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, -5000000000.0], 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) < -5e9 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 52.0

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

   5. Applied rewrites 52.0%

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

   if -5e9 < (/.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 93.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -5000000000:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 4.0 -2.0)))
   (if (<= x -5e-43) t_0 (if (<= x 5.2e+15) (fma (/ y z) -4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 4.0, -2.0);
	double tmp;
	if (x <= -5e-43) {
		tmp = t_0;
	} else if (x <= 5.2e+15) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 4.0, -2.0)
	tmp = 0.0
	if (x <= -5e-43)
		tmp = t_0;
	elseif (x <= 5.2e+15)
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]}, If[LessEqual[x, -5e-43], t$95$0, If[LessEqual[x, 5.2e+15], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 5.2e15 < x

   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 84.2

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

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

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

   if -5.00000000000000019e-43 < x < 5.2e15

   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 50.6

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

   5. Applied rewrites 50.6%

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

   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. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. associate-*r/ N/A

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

      14. div-sub N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-in N/A

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

      18. associate-/l* N/A

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

      19. *-inverses N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. mul-1-neg N/A

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

   8. Applied rewrites 92.6%

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

Alternative 6: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) 4.0 -2.0)))
   (if (<= x -5e-43) t_0 (if (<= x 5.2e+15) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), 4.0, -2.0);
	double tmp;
	if (x <= -5e-43) {
		tmp = t_0;
	} else if (x <= 5.2e+15) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), 4.0, -2.0)
	tmp = 0.0
	if (x <= -5e-43)
		tmp = t_0;
	elseif (x <= 5.2e+15)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]}, If[LessEqual[x, -5e-43], t$95$0, If[LessEqual[x, 5.2e+15], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 5.2e15 < x

   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 84.2

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

   5. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{4}{z}, x, -2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.3%

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

   if -5.00000000000000019e-43 < x < 5.2e15

   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 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.8%

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

Alternative 7: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ 4.0 z) x -2.0)))
   (if (<= x -5e-43) t_0 (if (<= x 5.2e+15) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((4.0 / z), x, -2.0);
	double tmp;
	if (x <= -5e-43) {
		tmp = t_0;
	} else if (x <= 5.2e+15) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(4.0 / z), x, -2.0)
	tmp = 0.0
	if (x <= -5e-43)
		tmp = t_0;
	elseif (x <= 5.2e+15)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]}, If[LessEqual[x, -5e-43], t$95$0, If[LessEqual[x, 5.2e+15], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 5.2e15 < x

   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 84.2

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

   5. Applied rewrites 84.2%

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

   if -5.00000000000000019e-43 < x < 5.2e15

   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 42.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 91.8%

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

Alternative 8: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -7.5e+85) t_0 (if (<= x 1.7e+27) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double tmp;
	if (x <= -7.5e+85) {
		tmp = t_0;
	} else if (x <= 1.7e+27) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	tmp = 0.0
	if (x <= -7.5e+85)
		tmp = t_0;
	elseif (x <= 1.7e+27)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -7.5e+85], t$95$0, If[LessEqual[x, 1.7e+27], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999942e85 or 1.7e27 < x

   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. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -7.49999999999999942e85 < x < 1.7e27

   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 41.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.7% 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 30.7%

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

Developer Target 1: 98.1% 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 2024332 
(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))