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

Percentage Accurate: 99.8% → 99.9%

Time: 5.6s

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: 99.8% 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: 99.9% accurate, 1.4× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. 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. Taylor expanded in y around 0 100.0%

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

   1. associate-*r/ 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 56.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y} \cdot -4\\ t_1 := 1 + \frac{4}{\frac{y}{x}}\\ \mathbf{if}\;x \leq -6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-288}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-208}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ z y) -4.0))) (t_1 (+ 1.0 (/ 4.0 (/ y x)))))
   (if (<= x -6e+18)
     t_1
     (if (<= x -7e-260)
       4.0
       (if (<= x 8e-288)
         t_0
         (if (<= x 1.2e-208)
           4.0
           (if (<= x 6.5e-58) t_0 (if (<= x 2e+27) 4.0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z / y) * -4.0);
	double t_1 = 1.0 + (4.0 / (y / x));
	double tmp;
	if (x <= -6e+18) {
		tmp = t_1;
	} else if (x <= -7e-260) {
		tmp = 4.0;
	} else if (x <= 8e-288) {
		tmp = t_0;
	} else if (x <= 1.2e-208) {
		tmp = 4.0;
	} else if (x <= 6.5e-58) {
		tmp = t_0;
	} else if (x <= 2e+27) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((z / y) * (-4.0d0))
    t_1 = 1.0d0 + (4.0d0 / (y / x))
    if (x <= (-6d+18)) then
        tmp = t_1
    else if (x <= (-7d-260)) then
        tmp = 4.0d0
    else if (x <= 8d-288) then
        tmp = t_0
    else if (x <= 1.2d-208) then
        tmp = 4.0d0
    else if (x <= 6.5d-58) then
        tmp = t_0
    else if (x <= 2d+27) then
        tmp = 4.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z / y) * -4.0);
	double t_1 = 1.0 + (4.0 / (y / x));
	double tmp;
	if (x <= -6e+18) {
		tmp = t_1;
	} else if (x <= -7e-260) {
		tmp = 4.0;
	} else if (x <= 8e-288) {
		tmp = t_0;
	} else if (x <= 1.2e-208) {
		tmp = 4.0;
	} else if (x <= 6.5e-58) {
		tmp = t_0;
	} else if (x <= 2e+27) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((z / y) * -4.0)
	t_1 = 1.0 + (4.0 / (y / x))
	tmp = 0
	if x <= -6e+18:
		tmp = t_1
	elif x <= -7e-260:
		tmp = 4.0
	elif x <= 8e-288:
		tmp = t_0
	elif x <= 1.2e-208:
		tmp = 4.0
	elif x <= 6.5e-58:
		tmp = t_0
	elif x <= 2e+27:
		tmp = 4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(z / y) * -4.0))
	t_1 = Float64(1.0 + Float64(4.0 / Float64(y / x)))
	tmp = 0.0
	if (x <= -6e+18)
		tmp = t_1;
	elseif (x <= -7e-260)
		tmp = 4.0;
	elseif (x <= 8e-288)
		tmp = t_0;
	elseif (x <= 1.2e-208)
		tmp = 4.0;
	elseif (x <= 6.5e-58)
		tmp = t_0;
	elseif (x <= 2e+27)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((z / y) * -4.0);
	t_1 = 1.0 + (4.0 / (y / x));
	tmp = 0.0;
	if (x <= -6e+18)
		tmp = t_1;
	elseif (x <= -7e-260)
		tmp = 4.0;
	elseif (x <= 8e-288)
		tmp = t_0;
	elseif (x <= 1.2e-208)
		tmp = 4.0;
	elseif (x <= 6.5e-58)
		tmp = t_0;
	elseif (x <= 2e+27)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(4.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6e+18], t$95$1, If[LessEqual[x, -7e-260], 4.0, If[LessEqual[x, 8e-288], t$95$0, If[LessEqual[x, 1.2e-208], 4.0, If[LessEqual[x, 6.5e-58], t$95$0, If[LessEqual[x, 2e+27], 4.0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6e18 or 2e27 < x

   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. +-commutative 99.7%

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

      3. 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. Step-by-step derivation

      1. fma-udef 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/r/ 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r- 99.6%

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

      6. fma-udef 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 68.3%

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

   if -6e18 < x < -6.9999999999999999e-260 or 8.00000000000000046e-288 < x < 1.1999999999999999e-208 or 6.49999999999999964e-58 < x < 2e27

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. 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. Taylor expanded in y around inf 61.4%

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

   if -6.9999999999999999e-260 < x < 8.00000000000000046e-288 or 1.1999999999999999e-208 < x < 6.49999999999999964e-58

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 62.6%

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

      1. *-commutative 62.6%

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

   6. Simplified 62.6%

      \[\leadsto 1 + \color{blue}{\frac{z}{y} \cdot -4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+18}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-288}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-208}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-58}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \end{array} \]

Alternative 3: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y} \cdot -4\\ t_1 := 1 + \frac{4 \cdot x}{y}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-292}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (/ z y) -4.0))) (t_1 (+ 1.0 (/ (* 4.0 x) y))))
   (if (<= x -1.75e+16)
     t_1
     (if (<= x -3e-260)
       4.0
       (if (<= x 2.5e-292)
         t_0
         (if (<= x 1.7e-208)
           4.0
           (if (<= x 8.8e-61) t_0 (if (<= x 1.95e+27) 4.0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z / y) * -4.0);
	double t_1 = 1.0 + ((4.0 * x) / y);
	double tmp;
	if (x <= -1.75e+16) {
		tmp = t_1;
	} else if (x <= -3e-260) {
		tmp = 4.0;
	} else if (x <= 2.5e-292) {
		tmp = t_0;
	} else if (x <= 1.7e-208) {
		tmp = 4.0;
	} else if (x <= 8.8e-61) {
		tmp = t_0;
	} else if (x <= 1.95e+27) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((z / y) * (-4.0d0))
    t_1 = 1.0d0 + ((4.0d0 * x) / y)
    if (x <= (-1.75d+16)) then
        tmp = t_1
    else if (x <= (-3d-260)) then
        tmp = 4.0d0
    else if (x <= 2.5d-292) then
        tmp = t_0
    else if (x <= 1.7d-208) then
        tmp = 4.0d0
    else if (x <= 8.8d-61) then
        tmp = t_0
    else if (x <= 1.95d+27) then
        tmp = 4.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((z / y) * -4.0);
	double t_1 = 1.0 + ((4.0 * x) / y);
	double tmp;
	if (x <= -1.75e+16) {
		tmp = t_1;
	} else if (x <= -3e-260) {
		tmp = 4.0;
	} else if (x <= 2.5e-292) {
		tmp = t_0;
	} else if (x <= 1.7e-208) {
		tmp = 4.0;
	} else if (x <= 8.8e-61) {
		tmp = t_0;
	} else if (x <= 1.95e+27) {
		tmp = 4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((z / y) * -4.0)
	t_1 = 1.0 + ((4.0 * x) / y)
	tmp = 0
	if x <= -1.75e+16:
		tmp = t_1
	elif x <= -3e-260:
		tmp = 4.0
	elif x <= 2.5e-292:
		tmp = t_0
	elif x <= 1.7e-208:
		tmp = 4.0
	elif x <= 8.8e-61:
		tmp = t_0
	elif x <= 1.95e+27:
		tmp = 4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(z / y) * -4.0))
	t_1 = Float64(1.0 + Float64(Float64(4.0 * x) / y))
	tmp = 0.0
	if (x <= -1.75e+16)
		tmp = t_1;
	elseif (x <= -3e-260)
		tmp = 4.0;
	elseif (x <= 2.5e-292)
		tmp = t_0;
	elseif (x <= 1.7e-208)
		tmp = 4.0;
	elseif (x <= 8.8e-61)
		tmp = t_0;
	elseif (x <= 1.95e+27)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((z / y) * -4.0);
	t_1 = 1.0 + ((4.0 * x) / y);
	tmp = 0.0;
	if (x <= -1.75e+16)
		tmp = t_1;
	elseif (x <= -3e-260)
		tmp = 4.0;
	elseif (x <= 2.5e-292)
		tmp = t_0;
	elseif (x <= 1.7e-208)
		tmp = 4.0;
	elseif (x <= 8.8e-61)
		tmp = t_0;
	elseif (x <= 1.95e+27)
		tmp = 4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+16], t$95$1, If[LessEqual[x, -3e-260], 4.0, If[LessEqual[x, 2.5e-292], t$95$0, If[LessEqual[x, 1.7e-208], 4.0, If[LessEqual[x, 8.8e-61], t$95$0, If[LessEqual[x, 1.95e+27], 4.0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75e16 or 1.9499999999999999e27 < x

   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. +-commutative 99.7%

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

      3. 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. Taylor expanded in x around inf 68.5%

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

      1. associate-*r/ 68.5%

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

      2. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   if -1.75e16 < x < -3.0000000000000001e-260 or 2.49999999999999991e-292 < x < 1.7e-208 or 8.80000000000000035e-61 < x < 1.9499999999999999e27

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. 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. Taylor expanded in y around inf 61.4%

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

   if -3.0000000000000001e-260 < x < 2.49999999999999991e-292 or 1.7e-208 < x < 8.80000000000000035e-61

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 62.6%

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

      1. *-commutative 62.6%

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

   6. Simplified 62.6%

      \[\leadsto 1 + \color{blue}{\frac{z}{y} \cdot -4} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-260}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-292}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-61}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+27}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{4 \cdot x}{y}\\ \end{array} \]

Alternative 4: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+193}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+130}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e+193) 4.0 (if (<= y 6e+130) (+ 1.0 (/ 4.0 (/ y (- x z)))) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+193) {
		tmp = 4.0;
	} else if (y <= 6e+130) {
		tmp = 1.0 + (4.0 / (y / (x - 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) :: tmp
    if (y <= (-6d+193)) then
        tmp = 4.0d0
    else if (y <= 6d+130) then
        tmp = 1.0d0 + (4.0d0 / (y / (x - z)))
    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 <= -6e+193) {
		tmp = 4.0;
	} else if (y <= 6e+130) {
		tmp = 1.0 + (4.0 / (y / (x - z)));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e+193:
		tmp = 4.0
	elif y <= 6e+130:
		tmp = 1.0 + (4.0 / (y / (x - z)))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e+193)
		tmp = 4.0;
	elseif (y <= 6e+130)
		tmp = Float64(1.0 + Float64(4.0 / Float64(y / Float64(x - z))));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6e+193)
		tmp = 4.0;
	elseif (y <= 6e+130)
		tmp = 1.0 + (4.0 / (y / (x - z)));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e+193], 4.0, If[LessEqual[y, 6e+130], N[(1.0 + N[(4.0 / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6e193 or 5.9999999999999999e130 < y

   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/ 99.6%

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

      2. +-commutative 99.6%

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

      3. 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. Taylor expanded in y around inf 80.2%

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

   if -6e193 < y < 5.9999999999999999e130

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r/ 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r- 99.7%

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

      6. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 81.2%

      \[\leadsto 1 + \frac{4}{\color{blue}{\frac{y}{x - z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+193}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+130}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 5: 55.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+41} \lor \neg \left(z \leq 0.00195\right):\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e+41) (not (<= z 0.00195))) (+ 1.0 (* (/ z y) -4.0)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+41) || !(z <= 0.00195)) {
		tmp = 1.0 + ((z / y) * -4.0);
	} 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 ((z <= (-1.15d+41)) .or. (.not. (z <= 0.00195d0))) then
        tmp = 1.0d0 + ((z / y) * (-4.0d0))
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e+41) || !(z <= 0.00195)) {
		tmp = 1.0 + ((z / y) * -4.0);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e+41) or not (z <= 0.00195):
		tmp = 1.0 + ((z / y) * -4.0)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e+41) || !(z <= 0.00195))
		tmp = Float64(1.0 + Float64(Float64(z / y) * -4.0));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e+41) || ~((z <= 0.00195)))
		tmp = 1.0 + ((z / y) * -4.0);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.15e+41], N[Not[LessEqual[z, 0.00195]], $MachinePrecision]], N[(1.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if z < -1.1499999999999999e41 or 0.0019499999999999999 < z

   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/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 66.2%

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

      1. *-commutative 66.2%

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

   6. Simplified 66.2%

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

   if -1.1499999999999999e41 < z < 0.0019499999999999999

   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. +-commutative 99.7%

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

      3. 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. Taylor expanded in y around inf 48.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+41} \lor \neg \left(z \leq 0.00195\right):\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 6: 7.6% 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 99.9%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. 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. Taylor expanded in z around inf 36.2%

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

   1. *-commutative 36.2%

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

6. Simplified 36.2%

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

7. Taylor expanded in z around 0 7.7%

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

8. Final simplification 7.7%

   \[\leadsto 1 \]

Alternative 7: 34.0% 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 99.9%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. 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. Taylor expanded in y around inf 35.7%

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

5. Final simplification 35.7%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023311 
(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)))