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

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

Alternatives: 7

Speedup: 1.0×

• 20.9% 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: 99.8% 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: 99.8% 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]

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. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(x - z\right)}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -1000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10000000000:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- x z)) y)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -1000000000000.0)
     t_0
     (if (<= t_1 10000000000.0) (fma 4.0 (/ x y) 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x - z)) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -1000000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 10000000000.0) {
		tmp = fma(4.0, (x / y), 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x - z)) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -1000000000000.0)
		tmp = t_0;
	elseif (t_1 <= 10000000000.0)
		tmp = fma(4.0, Float64(x / y), 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $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, -1000000000000.0], t$95$0, If[LessEqual[t$95$1, 10000000000.0], N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision], t$95$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) < -1e12 or 1e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   5. Simplified 98.6%

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

   if -1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 1e10

   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 z around 0

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

   5. Simplified 99.0%

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

Alternative 3: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) y)) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -100000000.0) t_0 (if (<= t_1 5.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} 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 * x) / y
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-100000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    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 * x) / y;
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * x) / y
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -100000000.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / y)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * x) / y;
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / y), $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, -100000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, t$95$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) < -1e8 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 55.1%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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. Simplified 97.7%

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

Alternative 4: 66.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{y}\\ t_1 := \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\\ \mathbf{if}\;t\_1 \leq -100000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 y))) (t_1 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))
   (if (<= t_1 -100000000.0) t_0 (if (<= t_1 5.0) 4.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (4.0d0 / y)
    t_1 = (4.0d0 * ((x + (y * 0.75d0)) - z)) / y
    if (t_1 <= (-100000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 5.0d0) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (4.0 / y);
	double t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	double tmp;
	if (t_1 <= -100000000.0) {
		tmp = t_0;
	} else if (t_1 <= 5.0) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / y)
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y
	tmp = 0
	if t_1 <= -100000000.0:
		tmp = t_0
	elif t_1 <= 5.0:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / y))
	t_1 = Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y)
	tmp = 0.0
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (4.0 / y);
	t_1 = (4.0 * ((x + (y * 0.75)) - z)) / y;
	tmp = 0.0;
	if (t_1 <= -100000000.0)
		tmp = t_0;
	elseif (t_1 <= 5.0)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / y), $MachinePrecision]), $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, -100000000.0], t$95$0, If[LessEqual[t$95$1, 5.0], 4.0, t$95$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) < -1e8 or 5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 55.1%

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

      1. associate-*l/ N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 55.0

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

   10. Applied egg-rr 55.0%

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

   if -1e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (+.f64 x (*.f64 y #s(literal 3/4 binary64))) z)) y) < 5

   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. Simplified 97.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq -100000000:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \leq 5:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(4, \frac{x}{y}, 4\right)\\ \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, -4, 4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma 4.0 (/ x y) 4.0)))
   (if (<= x -2.2) t_0 (if (<= x 5.2e-67) (fma (/ z y) -4.0 4.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(4.0, (x / y), 4.0);
	double tmp;
	if (x <= -2.2) {
		tmp = t_0;
	} else if (x <= 5.2e-67) {
		tmp = fma((z / y), -4.0, 4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(4.0, Float64(x / y), 4.0)
	tmp = 0.0
	if (x <= -2.2)
		tmp = t_0;
	elseif (x <= 5.2e-67)
		tmp = fma(Float64(z / y), -4.0, 4.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision] + 4.0), $MachinePrecision]}, If[LessEqual[x, -2.2], t$95$0, If[LessEqual[x, 5.2e-67], N[(N[(z / y), $MachinePrecision] * -4.0 + 4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002 or 5.1999999999999998e-67 < x

   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 0

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

   5. Simplified 87.4%

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

   if -2.2000000000000002 < x < 5.1999999999999998e-67

   1. Initial program 98.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

      9. sub-neg N/A

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

      10. distribute-rgt-in N/A

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

      11. metadata-eval N/A

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

      12. associate-+r+ N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. *-lft-identity N/A

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

      16. associate-*l/ N/A

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

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

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

   5. Simplified 93.8%

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

Alternative 6: 68.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(4.0 * N[(x / y), $MachinePrecision] + 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 z around 0

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

5. Simplified 70.2%

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

Alternative 7: 34.3% 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. Simplified 33.1%

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

Reproduce

herbie shell --seed 2024195 
(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)))