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

Percentage Accurate: 99.7% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.5×

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right) + 1} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{4 \cdot \frac{x}{y} + \left(4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1\right)} \]

   3. div-sub N/A

      \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} + 1\right) \]

   4. associate-/l* N/A

      \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) + 1\right) \]

   5. *-inverses N/A

      \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) + 1\right) \]

   6. metadata-eval N/A

      \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) + 1\right) \]

   7. +-commutative N/A

      \[\leadsto 4 \cdot \frac{x}{y} + \color{blue}{\left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{y} \cdot 4} + \left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \color{blue}{\left(\frac{3}{4} \cdot 4 + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \left(\color{blue}{3} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)\right) \]

   12. associate-+r+ N/A

       \[\leadsto \frac{x}{y} \cdot 4 + \color{blue}{\left(\left(1 + 3\right) + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)} \]

   13. metadata-eval N/A

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

   14. distribute-rgt1-in N/A

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

   15. distribute-rgt-in N/A

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

   16. associate-+l+ N/A

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

5. Applied rewrites 100.0%

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

Alternative 2: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\frac{-4 \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -20000.0)
     t_0
     (if (<= t_1 50000.0) 4.0 (if (<= t_1 4e+83) (/ (* -4.0 z) y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * 4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = 4.0;
	} else if (t_1 <= 4e+83) {
		tmp = (-4.0 * z) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) * 4.0d0
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 50000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 4d+83) then
        tmp = ((-4.0d0) * z) / y
    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 / y) * 4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = 4.0;
	} else if (t_1 <= 4e+83) {
		tmp = (-4.0 * z) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * 4.0
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 50000.0:
		tmp = 4.0
	elif t_1 <= 4e+83:
		tmp = (-4.0 * z) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * 4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = 4.0;
	elseif (t_1 <= 4e+83)
		tmp = Float64(Float64(-4.0 * z) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / y) * 4.0;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = 4.0;
	elseif (t_1 <= 4e+83)
		tmp = (-4.0 * z) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 50000.0], 4.0, If[LessEqual[t$95$1, 4e+83], N[(N[(-4.0 * z), $MachinePrecision] / y), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 4.00000000000000012e83 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e4

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{4} \]

   if 5e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 4.00000000000000012e83

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.6%

      \[\leadsto \frac{-4 \cdot z}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 66.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 4\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 50000:\\ \;\;\;\;4\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+83}:\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ x y) 4.0)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -20000.0)
     t_0
     (if (<= t_1 50000.0) 4.0 (if (<= t_1 4e+83) (* (/ -4.0 y) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (x / y) * 4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = 4.0;
	} else if (t_1 <= 4e+83) {
		tmp = (-4.0 / y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y) * 4.0d0
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-20000.0d0)) then
        tmp = t_0
    else if (t_1 <= 50000.0d0) then
        tmp = 4.0d0
    else if (t_1 <= 4d+83) then
        tmp = ((-4.0d0) / y) * z
    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 / y) * 4.0;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = t_0;
	} else if (t_1 <= 50000.0) {
		tmp = 4.0;
	} else if (t_1 <= 4e+83) {
		tmp = (-4.0 / y) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / y) * 4.0
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -20000.0:
		tmp = t_0
	elif t_1 <= 50000.0:
		tmp = 4.0
	elif t_1 <= 4e+83:
		tmp = (-4.0 / y) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / y) * 4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = 4.0;
	elseif (t_1 <= 4e+83)
		tmp = Float64(Float64(-4.0 / y) * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / y) * 4.0;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -20000.0)
		tmp = t_0;
	elseif (t_1 <= 50000.0)
		tmp = 4.0;
	elseif (t_1 <= 4e+83)
		tmp = (-4.0 / y) * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], t$95$0, If[LessEqual[t$95$1, 50000.0], 4.0, If[LessEqual[t$95$1, 4e+83], N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 4.00000000000000012e83 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e4

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

      \[\leadsto \color{blue}{4} \]

   if 5e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 4.00000000000000012e83

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 67.2

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

   5. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{-4}{y} \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+16} \lor \neg \left(t\_0 \leq 50000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -5e+16) (not (<= t_0 50000.0)))
     (* (/ (- x z) y) 4.0)
     (fma (/ 4.0 y) x 4.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -5e+16) || !(t_0 <= 50000.0)) {
		tmp = ((x - z) / y) * 4.0;
	} else {
		tmp = fma((4.0 / y), x, 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -5e+16) || !(t_0 <= 50000.0))
		tmp = Float64(Float64(Float64(x - z) / y) * 4.0);
	else
		tmp = fma(Float64(4.0 / y), x, 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+16], N[Not[LessEqual[t$95$0, 50000.0]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -5e16 or 5e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if -5e16 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e4

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{y} + \left(4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1\right)} \]

      3. div-sub N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} + 1\right) \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) + 1\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) + 1\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) + 1\right) \]

      7. +-commutative N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \color{blue}{\left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot 4} + \left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \color{blue}{\left(\frac{3}{4} \cdot 4 + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \left(\color{blue}{3} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \color{blue}{\left(\left(1 + 3\right) + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)} \]

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{-1 \cdot x}{y}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{-1 \cdot x}{y}\right) \]

      7. *-inverses N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{-1 \cdot x}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{-1 \cdot x}{y}\right) \]

      9. associate-*l/ N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\frac{-1}{y} \cdot x}\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y} \cdot x\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)} \cdot x\right) \]

      12. unsub-neg N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right) \cdot x\right)\right)\right)} \]

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

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) \cdot x}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{1}{y}} \cdot x\right) \]

      15. associate-*l/ N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{1 \cdot x}{y}}\right) \]

      16. *-lft-identity N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \frac{\color{blue}{x}}{y}\right) \]

      17. distribute-lft-in N/A

          \[\leadsto 1 + \color{blue}{\left(4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{y}\right)} \]

      18. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + 4 \cdot \frac{3}{4}\right) + 4 \cdot \frac{x}{y}} \]

   8. Applied rewrites 99.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -5 \cdot 10^{+16} \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 50000\right):\\ \;\;\;\;\frac{x - z}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_0 \leq -20000 \lor \neg \left(t\_0 \leq 50000\right):\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (or (<= t_0 -20000.0) (not (<= t_0 50000.0))) (* (/ -4.0 y) z) 4.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 50000.0)) {
		tmp = (-4.0 / y) * z;
	} else {
		tmp = 4.0;
	}
	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) :: tmp
    t_0 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if ((t_0 <= (-20000.0d0)) .or. (.not. (t_0 <= 50000.0d0))) then
        tmp = ((-4.0d0) / y) * z
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if ((t_0 <= -20000.0) || !(t_0 <= 50000.0)) {
		tmp = (-4.0 / y) * z;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if (t_0 <= -20000.0) or not (t_0 <= 50000.0):
		tmp = (-4.0 / y) * z
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if ((t_0 <= -20000.0) || !(t_0 <= 50000.0))
		tmp = Float64(Float64(-4.0 / y) * z);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if ((t_0 <= -20000.0) || ~((t_0 <= 50000.0)))
		tmp = (-4.0 / y) * z;
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000.0], N[Not[LessEqual[t$95$0, 50000.0]], $MachinePrecision]], N[(N[(-4.0 / y), $MachinePrecision] * z), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < -2e4 or 5e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 47.6

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

   5. Applied rewrites 47.6%

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

   if -2e4 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5e4

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -20000 \lor \neg \left(\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 50000\right):\\ \;\;\;\;\frac{-4}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+86} \lor \neg \left(x \leq 1.35 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.8e+86) (not (<= x 1.35e+31)))
   (fma (/ 4.0 y) x 4.0)
   (fma -4.0 (/ z y) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.8e+86) || !(x <= 1.35e+31)) {
		tmp = fma((4.0 / y), x, 4.0);
	} else {
		tmp = fma(-4.0, (z / y), 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.8e+86) || !(x <= 1.35e+31))
		tmp = fma(Float64(4.0 / y), x, 4.0);
	else
		tmp = fma(-4.0, Float64(z / y), 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.8e+86], N[Not[LessEqual[x, 1.35e+31]], $MachinePrecision]], N[(N[(4.0 / y), $MachinePrecision] * x + 4.0), $MachinePrecision], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000004e86 or 1.34999999999999993e31 < x

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}\right) + 1} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{4 \cdot \frac{x}{y} + \left(4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1\right)} \]

      3. div-sub N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} + 1\right) \]

      4. associate-/l* N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) + 1\right) \]

      5. *-inverses N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) + 1\right) \]

      6. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \left(4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) + 1\right) \]

      7. +-commutative N/A

         \[\leadsto 4 \cdot \frac{x}{y} + \color{blue}{\left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot 4} + \left(1 + 4 \cdot \left(\frac{3}{4} - \frac{z}{y}\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \color{blue}{\left(\frac{3}{4} \cdot 4 + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)}\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \left(1 + \left(\color{blue}{3} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto \frac{x}{y} \cdot 4 + \color{blue}{\left(\left(1 + 3\right) + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4\right)} \]

      13. metadata-eval N/A

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

      14. distribute-rgt1-in N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-+l+ N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \frac{3}{4} \cdot y}{y}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y + x}}{y} \]

      2. remove-double-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)}}{y} \]

      3. mul-1-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\frac{3}{4} \cdot y + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot x}\right)\right)}{y} \]

      4. unsub-neg N/A

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{\frac{3}{4} \cdot y - -1 \cdot x}}{y} \]

      5. div-sub N/A

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{-1 \cdot x}{y}\right)} \]

      6. associate-*r/ N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{-1 \cdot x}{y}\right) \]

      7. *-inverses N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{-1 \cdot x}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{-1 \cdot x}{y}\right) \]

      9. associate-*l/ N/A

         \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\frac{-1}{y} \cdot x}\right) \]

      10. metadata-eval N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y} \cdot x\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)} \cdot x\right) \]

      12. unsub-neg N/A

          \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right) \cdot x\right)\right)\right)} \]

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

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) \cdot x}\right) \]

      14. remove-double-neg N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{1}{y}} \cdot x\right) \]

      15. associate-*l/ N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \color{blue}{\frac{1 \cdot x}{y}}\right) \]

      16. *-lft-identity N/A

          \[\leadsto 1 + 4 \cdot \left(\frac{3}{4} + \frac{\color{blue}{x}}{y}\right) \]

      17. distribute-lft-in N/A

          \[\leadsto 1 + \color{blue}{\left(4 \cdot \frac{3}{4} + 4 \cdot \frac{x}{y}\right)} \]

      18. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(1 + 4 \cdot \frac{3}{4}\right) + 4 \cdot \frac{x}{y}} \]

   8. Applied rewrites 90.5%

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

   if -2.80000000000000004e86 < x < 1.34999999999999993e31

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} + 1 \]

      3. associate-/l* N/A

         \[\leadsto 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) + 1 \]

      4. *-inverses N/A

         \[\leadsto 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) + 1 \]

      5. metadata-eval N/A

         \[\leadsto 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) + 1 \]

      6. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      7. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      10. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      11. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \color{blue}{3}\right) + 1 \]

      15. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \left(3 + 1\right)} \]

      16. metadata-eval N/A

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

   5. Applied rewrites 90.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+86} \lor \neg \left(x \leq 1.35 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{y}, x, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+108} \lor \neg \left(x \leq 4 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.12e+108) (not (<= x 4e+199)))
   (* (/ x y) 4.0)
   (fma -4.0 (/ z y) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e+108) || !(x <= 4e+199)) {
		tmp = (x / y) * 4.0;
	} else {
		tmp = fma(-4.0, (z / y), 4.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.12e+108) || !(x <= 4e+199))
		tmp = Float64(Float64(x / y) * 4.0);
	else
		tmp = fma(-4.0, Float64(z / y), 4.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.12e+108], N[Not[LessEqual[x, 4e+199]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * 4.0), $MachinePrecision], N[(-4.0 * N[(z / y), $MachinePrecision] + 4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.11999999999999994e108 or 4.00000000000000039e199 < x

   1. Initial program 98.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.11999999999999994e108 < x < 4.00000000000000039e199

   1. Initial program 99.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{\frac{3}{4} \cdot y - z}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{4 \cdot \frac{\frac{3}{4} \cdot y - z}{y} + 1} \]

      2. div-sub N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{\frac{3}{4} \cdot y}{y} - \frac{z}{y}\right)} + 1 \]

      3. associate-/l* N/A

         \[\leadsto 4 \cdot \left(\color{blue}{\frac{3}{4} \cdot \frac{y}{y}} - \frac{z}{y}\right) + 1 \]

      4. *-inverses N/A

         \[\leadsto 4 \cdot \left(\frac{3}{4} \cdot \color{blue}{1} - \frac{z}{y}\right) + 1 \]

      5. metadata-eval N/A

         \[\leadsto 4 \cdot \left(\color{blue}{\frac{3}{4}} - \frac{z}{y}\right) + 1 \]

      6. sub-neg N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\frac{3}{4} + \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right)} + 1 \]

      7. +-commutative N/A

         \[\leadsto 4 \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) + \frac{3}{4}\right)} + 1 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot 4 + \frac{3}{4} \cdot 4\right)} + 1 \]

      9. *-commutative N/A

         \[\leadsto \left(\color{blue}{4 \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      10. *-lft-identity N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{1 \cdot z}}{y}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      11. associate-*l/ N/A

          \[\leadsto \left(4 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{y} \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

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

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(4 \cdot \left(\frac{1}{y} \cdot z\right)\right)\right)} + \frac{3}{4} \cdot 4\right) + 1 \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(4 \cdot \frac{1}{y}\right) \cdot z}\right)\right) + \frac{3}{4} \cdot 4\right) + 1 \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \color{blue}{3}\right) + 1 \]

      15. associate-+l+ N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(4 \cdot \frac{1}{y}\right) \cdot z\right)\right) + \left(3 + 1\right)} \]

      16. metadata-eval N/A

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

   5. Applied rewrites 85.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+108} \lor \neg \left(x \leq 4 \cdot 10^{+199}\right):\\ \;\;\;\;\frac{x}{y} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{z}{y}, 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.6% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.2%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 38.0%

   \[\leadsto \color{blue}{4} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))