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

Percentage Accurate: 99.7% → 100.0%

Time: 7.2s

Alternatives: 7

Speedup: 1.4×

• 21.1% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ 4 + 4 \cdot \frac{x - z}{y} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.3%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

   3. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

   4. +-commutative 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   5. fma-def 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 100.0%

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

6. Final simplification 100.0%

   \[\leadsto 4 + 4 \cdot \frac{x - z}{y} \]
7. Add Preprocessing

Alternative 2: 53.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{z \cdot -4}{y}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-60}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-215}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-253}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (/ (* z -4.0) y)))
   (if (<= x -1.65e+50)
     t_0
     (if (<= x -3.1e-18)
       t_1
       (if (<= x -1.05e-42)
         t_0
         (if (<= x -2.2e-60)
           4.0
           (if (<= x -1.45e-174)
             t_1
             (if (<= x -1.45e-215)
               4.0
               (if (<= x 1.55e-299)
                 t_1
                 (if (<= x 2.2e-253) 4.0 (if (<= x 1.65e+95) t_1 t_0)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (z * -4.0) / y;
	double tmp;
	if (x <= -1.65e+50) {
		tmp = t_0;
	} else if (x <= -3.1e-18) {
		tmp = t_1;
	} else if (x <= -1.05e-42) {
		tmp = t_0;
	} else if (x <= -2.2e-60) {
		tmp = 4.0;
	} else if (x <= -1.45e-174) {
		tmp = t_1;
	} else if (x <= -1.45e-215) {
		tmp = 4.0;
	} else if (x <= 1.55e-299) {
		tmp = t_1;
	} else if (x <= 2.2e-253) {
		tmp = 4.0;
	} else if (x <= 1.65e+95) {
		tmp = t_1;
	} 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 = (z * (-4.0d0)) / y
    if (x <= (-1.65d+50)) then
        tmp = t_0
    else if (x <= (-3.1d-18)) then
        tmp = t_1
    else if (x <= (-1.05d-42)) then
        tmp = t_0
    else if (x <= (-2.2d-60)) then
        tmp = 4.0d0
    else if (x <= (-1.45d-174)) then
        tmp = t_1
    else if (x <= (-1.45d-215)) then
        tmp = 4.0d0
    else if (x <= 1.55d-299) then
        tmp = t_1
    else if (x <= 2.2d-253) then
        tmp = 4.0d0
    else if (x <= 1.65d+95) then
        tmp = t_1
    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 = (z * -4.0) / y;
	double tmp;
	if (x <= -1.65e+50) {
		tmp = t_0;
	} else if (x <= -3.1e-18) {
		tmp = t_1;
	} else if (x <= -1.05e-42) {
		tmp = t_0;
	} else if (x <= -2.2e-60) {
		tmp = 4.0;
	} else if (x <= -1.45e-174) {
		tmp = t_1;
	} else if (x <= -1.45e-215) {
		tmp = 4.0;
	} else if (x <= 1.55e-299) {
		tmp = t_1;
	} else if (x <= 2.2e-253) {
		tmp = 4.0;
	} else if (x <= 1.65e+95) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = (z * -4.0) / y
	tmp = 0
	if x <= -1.65e+50:
		tmp = t_0
	elif x <= -3.1e-18:
		tmp = t_1
	elif x <= -1.05e-42:
		tmp = t_0
	elif x <= -2.2e-60:
		tmp = 4.0
	elif x <= -1.45e-174:
		tmp = t_1
	elif x <= -1.45e-215:
		tmp = 4.0
	elif x <= 1.55e-299:
		tmp = t_1
	elif x <= 2.2e-253:
		tmp = 4.0
	elif x <= 1.65e+95:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(Float64(z * -4.0) / y)
	tmp = 0.0
	if (x <= -1.65e+50)
		tmp = t_0;
	elseif (x <= -3.1e-18)
		tmp = t_1;
	elseif (x <= -1.05e-42)
		tmp = t_0;
	elseif (x <= -2.2e-60)
		tmp = 4.0;
	elseif (x <= -1.45e-174)
		tmp = t_1;
	elseif (x <= -1.45e-215)
		tmp = 4.0;
	elseif (x <= 1.55e-299)
		tmp = t_1;
	elseif (x <= 2.2e-253)
		tmp = 4.0;
	elseif (x <= 1.65e+95)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = (z * -4.0) / y;
	tmp = 0.0;
	if (x <= -1.65e+50)
		tmp = t_0;
	elseif (x <= -3.1e-18)
		tmp = t_1;
	elseif (x <= -1.05e-42)
		tmp = t_0;
	elseif (x <= -2.2e-60)
		tmp = 4.0;
	elseif (x <= -1.45e-174)
		tmp = t_1;
	elseif (x <= -1.45e-215)
		tmp = 4.0;
	elseif (x <= 1.55e-299)
		tmp = t_1;
	elseif (x <= 2.2e-253)
		tmp = 4.0;
	elseif (x <= 1.65e+95)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -1.65e+50], t$95$0, If[LessEqual[x, -3.1e-18], t$95$1, If[LessEqual[x, -1.05e-42], t$95$0, If[LessEqual[x, -2.2e-60], 4.0, If[LessEqual[x, -1.45e-174], t$95$1, If[LessEqual[x, -1.45e-215], 4.0, If[LessEqual[x, 1.55e-299], t$95$1, If[LessEqual[x, 2.2e-253], 4.0, If[LessEqual[x, 1.65e+95], t$95$1, t$95$0]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e50 or -3.10000000000000007e-18 < x < -1.05000000000000003e-42 or 1.6499999999999999e95 < x

   1. Initial program 99.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.9%

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

   6. Taylor expanded in x around inf 74.9%

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

   if -1.65e50 < x < -3.10000000000000007e-18 or -2.1999999999999999e-60 < x < -1.45000000000000005e-174 or -1.45e-215 < x < 1.55e-299 or 2.19999999999999996e-253 < x < 1.6499999999999999e95

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.0%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in z around inf 59.3%

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

      1. associate-*r/ 59.3%

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

   8. Simplified 59.3%

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

   if -1.05000000000000003e-42 < x < -2.1999999999999999e-60 or -1.45000000000000005e-174 < x < -1.45e-215 or 1.55e-299 < x < 2.19999999999999996e-253

   1. Initial program 99.8%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 82.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-42}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-60}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-174}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-215}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-299}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-253}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+95}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-46} \lor \neg \left(z \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e-46) (not (<= z 4.8e+53)))
   (* (- x z) (/ 4.0 y))
   (+ 4.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e-46) || !(z <= 4.8e+53)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	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) :: tmp
    if ((z <= (-1.75d-46)) .or. (.not. (z <= 4.8d+53))) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e-46) || !(z <= 4.8e+53)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e-46) or not (z <= 4.8e+53):
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e-46) || !(z <= 4.8e+53))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e-46) || ~((z <= 4.8e+53)))
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e-46], N[Not[LessEqual[z, 4.8e+53]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7500000000000001e-46 or 4.8e53 < z

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in y around 0 90.5%

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

      1. associate-*r/ 89.7%

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

      2. associate-*l/ 90.3%

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

      3. *-commutative 90.3%

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

   8. Simplified 90.3%

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

   if -1.7500000000000001e-46 < z < 4.8e53

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.0%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 98.9%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in z around 0 90.0%

      \[\leadsto \color{blue}{4 + 4 \cdot \frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-46} \lor \neg \left(z \leq 4.8 \cdot 10^{+53}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+182}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.2e+182) 4.0 (if (<= y 1.85e+151) (* (- x z) (/ 4.0 y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.2e+182) {
		tmp = 4.0;
	} else if (y <= 1.85e+151) {
		tmp = (x - z) * (4.0 / y);
	} 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) :: tmp
    if (y <= (-6.2d+182)) then
        tmp = 4.0d0
    else if (y <= 1.85d+151) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.2e+182) {
		tmp = 4.0;
	} else if (y <= 1.85e+151) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.2e+182:
		tmp = 4.0
	elif y <= 1.85e+151:
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.2e+182)
		tmp = 4.0;
	elseif (y <= 1.85e+151)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.2e+182)
		tmp = 4.0;
	elseif (y <= 1.85e+151)
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.2e+182], 4.0, If[LessEqual[y, 1.85e+151], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999993e182 or 1.8499999999999999e151 < y

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.5%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 71.7%

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

   if -6.19999999999999993e182 < y < 1.8499999999999999e151

   1. Initial program 99.5%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.3%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.3%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.3%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.3%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.2%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in y around 0 81.0%

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

      1. associate-*r/ 80.5%

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

      2. associate-*l/ 80.3%

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

      3. *-commutative 80.3%

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

   8. Simplified 80.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+182}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-43} \lor \neg \left(x \leq 1.75 \cdot 10^{-49}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.2e-43) (not (<= x 1.75e-49))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.2e-43) || !(x <= 1.75e-49)) {
		tmp = 4.0 * (x / y);
	} 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) :: tmp
    if ((x <= (-8.2d-43)) .or. (.not. (x <= 1.75d-49))) then
        tmp = 4.0d0 * (x / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.2e-43) || !(x <= 1.75e-49)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.2e-43) or not (x <= 1.75e-49):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.2e-43) || !(x <= 1.75e-49))
		tmp = Float64(4.0 * Float64(x / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.2e-43) || ~((x <= 1.75e-49)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.2e-43], N[Not[LessEqual[x, 1.75e-49]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -8.1999999999999996e-43 or 1.75000000000000003e-49 < x

   1. Initial program 99.3%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.0%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.0%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around inf 60.9%

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

   if -8.1999999999999996e-43 < x < 1.75000000000000003e-49

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

      2. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

      3. sub-neg 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

      4. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

      5. fma-def 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 46.1%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-43} \lor \neg \left(x \leq 1.75 \cdot 10^{-49}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 7.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.3%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

   3. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

   4. +-commutative 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   5. fma-def 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 40.6%

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

6. Taylor expanded in x around 0 6.7%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 6.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 7: 34.9% accurate, 13.0× 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 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-*l/ 99.3%

      \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.75\right) - z\right)} \]

   2. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) + \left(-z\right)\right)} \]

   3. sub-neg 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(\left(x + y \cdot 0.75\right) - z\right)} \]

   4. +-commutative 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.75 + x\right)} - z\right) \]

   5. fma-def 99.3%

      \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\mathsf{fma}\left(y, 0.75, x\right)} - z\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{1 + \frac{4}{y} \cdot \left(\mathsf{fma}\left(y, 0.75, x\right) - z\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 28.1%

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

6. Final simplification 28.1%

   \[\leadsto 4 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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)))