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

Percentage Accurate: 99.8% → 99.9%

Time: 5.2s

Alternatives: 6

Speedup: 1.4×

• 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: 99.8% 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/ 99.7%

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

   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 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: 57.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y} \cdot -4\\ t_1 := 1 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-291}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-215}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+51}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 -2.8e-10)
     t_1
     (if (<= x -2e-256)
       t_0
       (if (<= x -3.5e-291)
         4.0
         (if (<= x 8.2e-265)
           t_0
           (if (<= x 1.6e-215)
             4.0
             (if (<= x 1.06e+24)
               t_0
               (if (<= x 1.55e+51) 4.0 (if (<= x 6e+52) t_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 <= -2.8e-10) {
		tmp = t_1;
	} else if (x <= -2e-256) {
		tmp = t_0;
	} else if (x <= -3.5e-291) {
		tmp = 4.0;
	} else if (x <= 8.2e-265) {
		tmp = t_0;
	} else if (x <= 1.6e-215) {
		tmp = 4.0;
	} else if (x <= 1.06e+24) {
		tmp = t_0;
	} else if (x <= 1.55e+51) {
		tmp = 4.0;
	} else if (x <= 6e+52) {
		tmp = t_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 <= (-2.8d-10)) then
        tmp = t_1
    else if (x <= (-2d-256)) then
        tmp = t_0
    else if (x <= (-3.5d-291)) then
        tmp = 4.0d0
    else if (x <= 8.2d-265) then
        tmp = t_0
    else if (x <= 1.6d-215) then
        tmp = 4.0d0
    else if (x <= 1.06d+24) then
        tmp = t_0
    else if (x <= 1.55d+51) then
        tmp = 4.0d0
    else if (x <= 6d+52) then
        tmp = t_0
    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 <= -2.8e-10) {
		tmp = t_1;
	} else if (x <= -2e-256) {
		tmp = t_0;
	} else if (x <= -3.5e-291) {
		tmp = 4.0;
	} else if (x <= 8.2e-265) {
		tmp = t_0;
	} else if (x <= 1.6e-215) {
		tmp = 4.0;
	} else if (x <= 1.06e+24) {
		tmp = t_0;
	} else if (x <= 1.55e+51) {
		tmp = 4.0;
	} else if (x <= 6e+52) {
		tmp = t_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 <= -2.8e-10:
		tmp = t_1
	elif x <= -2e-256:
		tmp = t_0
	elif x <= -3.5e-291:
		tmp = 4.0
	elif x <= 8.2e-265:
		tmp = t_0
	elif x <= 1.6e-215:
		tmp = 4.0
	elif x <= 1.06e+24:
		tmp = t_0
	elif x <= 1.55e+51:
		tmp = 4.0
	elif x <= 6e+52:
		tmp = t_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(x / y)))
	tmp = 0.0
	if (x <= -2.8e-10)
		tmp = t_1;
	elseif (x <= -2e-256)
		tmp = t_0;
	elseif (x <= -3.5e-291)
		tmp = 4.0;
	elseif (x <= 8.2e-265)
		tmp = t_0;
	elseif (x <= 1.6e-215)
		tmp = 4.0;
	elseif (x <= 1.06e+24)
		tmp = t_0;
	elseif (x <= 1.55e+51)
		tmp = 4.0;
	elseif (x <= 6e+52)
		tmp = t_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 <= -2.8e-10)
		tmp = t_1;
	elseif (x <= -2e-256)
		tmp = t_0;
	elseif (x <= -3.5e-291)
		tmp = 4.0;
	elseif (x <= 8.2e-265)
		tmp = t_0;
	elseif (x <= 1.6e-215)
		tmp = 4.0;
	elseif (x <= 1.06e+24)
		tmp = t_0;
	elseif (x <= 1.55e+51)
		tmp = 4.0;
	elseif (x <= 6e+52)
		tmp = t_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[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e-10], t$95$1, If[LessEqual[x, -2e-256], t$95$0, If[LessEqual[x, -3.5e-291], 4.0, If[LessEqual[x, 8.2e-265], t$95$0, If[LessEqual[x, 1.6e-215], 4.0, If[LessEqual[x, 1.06e+24], t$95$0, If[LessEqual[x, 1.55e+51], 4.0, If[LessEqual[x, 6e+52], t$95$0, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000015e-10 or 6e52 < 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 70.9%

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

   if -2.80000000000000015e-10 < x < -1.99999999999999995e-256 or -3.49999999999999996e-291 < x < 8.2e-265 or 1.6000000000000001e-215 < x < 1.06e24 or 1.55000000000000006e51 < x < 6e52

   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 65.2%

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

      1. *-commutative 65.2%

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

   6. Simplified 65.2%

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

   if -1.99999999999999995e-256 < x < -3.49999999999999996e-291 or 8.2e-265 < x < 1.6000000000000001e-215 or 1.06e24 < x < 1.55000000000000006e51

   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/ 99.8%

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

      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 y around inf 83.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-10}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-256}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-291}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-265}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-215}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+24}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+51}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+52}:\\ \;\;\;\;1 + \frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 54.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+75) 4.0 (if (<= y 3.5e+68) (+ 1.0 (* 4.0 (/ x y))) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+75) {
		tmp = 4.0;
	} else if (y <= 3.5e+68) {
		tmp = 1.0 + (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 (y <= (-1.4d+75)) then
        tmp = 4.0d0
    else if (y <= 3.5d+68) then
        tmp = 1.0d0 + (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 (y <= -1.4e+75) {
		tmp = 4.0;
	} else if (y <= 3.5e+68) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+75:
		tmp = 4.0
	elif y <= 3.5e+68:
		tmp = 1.0 + (4.0 * (x / y))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+75)
		tmp = 4.0;
	elseif (y <= 3.5e+68)
		tmp = Float64(1.0 + 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 (y <= -1.4e+75)
		tmp = 4.0;
	elseif (y <= 3.5e+68)
		tmp = 1.0 + (4.0 * (x / y));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+75], 4.0, If[LessEqual[y, 3.5e+68], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e75 or 3.49999999999999977e68 < 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/ 99.7%

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

      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 69.6%

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

   if -1.40000000000000006e75 < y < 3.49999999999999977e68

   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/ 99.7%

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

      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 49.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+75}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 4: 53.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+74}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{+68}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7e+74) 4.0 (if (<= y 3.25e+68) (* 4.0 (/ x y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7e+74) {
		tmp = 4.0;
	} else if (y <= 3.25e+68) {
		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 (y <= (-7d+74)) then
        tmp = 4.0d0
    else if (y <= 3.25d+68) 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 (y <= -7e+74) {
		tmp = 4.0;
	} else if (y <= 3.25e+68) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7e+74:
		tmp = 4.0
	elif y <= 3.25e+68:
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7e+74)
		tmp = 4.0;
	elseif (y <= 3.25e+68)
		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 (y <= -7e+74)
		tmp = 4.0;
	elseif (y <= 3.25e+68)
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7e+74], 4.0, If[LessEqual[y, 3.25e+68], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.00000000000000029e74 or 3.2500000000000002e68 < 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/ 99.7%

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

      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 69.6%

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

   if -7.00000000000000029e74 < y < 3.2500000000000002e68

   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/ 99.7%

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

      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 49.7%

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

   5. Taylor expanded in x around inf 48.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+74}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{+68}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 5: 7.5% 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/ 99.7%

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

   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 40.8%

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

5. Taylor expanded in x around 0 7.5%

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

6. Final simplification 7.5%

   \[\leadsto 1 \]

Alternative 6: 33.5% 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/ 99.7%

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

   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 33.1%

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

5. Final simplification 33.1%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023316 
(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)))