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

Percentage Accurate: 99.8% → 100.0%

Time: 8.7s

Alternatives: 10

Speedup: 1.2×

• 21.4% 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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left(4 \cdot \frac{x - z}{y} + 3\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 1.0 (+ (* 4.0 (/ (- x z) y)) 3.0)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x - z) / y)) + 3.0);
}

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 + ((4.0d0 * ((x - z) / y)) + 3.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x - z) / y)) + 3.0);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x - z) / y)) + 3.0)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x - z) / y)) + 3.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x - z) / y)) + 3.0);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

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 y around 0 100.0%

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

   1. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 57.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot \frac{-4}{y}\\ t_1 := 1 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-279}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 10^{-269}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-213}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z (/ -4.0 y)))) (t_1 (+ 1.0 (* 4.0 (/ x y)))))
   (if (<= x -3e+81)
     t_1
     (if (<= x -1.25e-195)
       t_0
       (if (<= x -4.5e-279)
         4.0
         (if (<= x 1e-269)
           t_0
           (if (<= x 2.3e-213) 4.0 (if (<= x 2.7e+44) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * (-4.0 / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -3e+81) {
		tmp = t_1;
	} else if (x <= -1.25e-195) {
		tmp = t_0;
	} else if (x <= -4.5e-279) {
		tmp = 4.0;
	} else if (x <= 1e-269) {
		tmp = t_0;
	} else if (x <= 2.3e-213) {
		tmp = 4.0;
	} else if (x <= 2.7e+44) {
		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 * ((-4.0d0) / y))
    t_1 = 1.0d0 + (4.0d0 * (x / y))
    if (x <= (-3d+81)) then
        tmp = t_1
    else if (x <= (-1.25d-195)) then
        tmp = t_0
    else if (x <= (-4.5d-279)) then
        tmp = 4.0d0
    else if (x <= 1d-269) then
        tmp = t_0
    else if (x <= 2.3d-213) then
        tmp = 4.0d0
    else if (x <= 2.7d+44) 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 * (-4.0 / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -3e+81) {
		tmp = t_1;
	} else if (x <= -1.25e-195) {
		tmp = t_0;
	} else if (x <= -4.5e-279) {
		tmp = 4.0;
	} else if (x <= 1e-269) {
		tmp = t_0;
	} else if (x <= 2.3e-213) {
		tmp = 4.0;
	} else if (x <= 2.7e+44) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * (-4.0 / y))
	t_1 = 1.0 + (4.0 * (x / y))
	tmp = 0
	if x <= -3e+81:
		tmp = t_1
	elif x <= -1.25e-195:
		tmp = t_0
	elif x <= -4.5e-279:
		tmp = 4.0
	elif x <= 1e-269:
		tmp = t_0
	elif x <= 2.3e-213:
		tmp = 4.0
	elif x <= 2.7e+44:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * Float64(-4.0 / y)))
	t_1 = Float64(1.0 + Float64(4.0 * Float64(x / y)))
	tmp = 0.0
	if (x <= -3e+81)
		tmp = t_1;
	elseif (x <= -1.25e-195)
		tmp = t_0;
	elseif (x <= -4.5e-279)
		tmp = 4.0;
	elseif (x <= 1e-269)
		tmp = t_0;
	elseif (x <= 2.3e-213)
		tmp = 4.0;
	elseif (x <= 2.7e+44)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * (-4.0 / y));
	t_1 = 1.0 + (4.0 * (x / y));
	tmp = 0.0;
	if (x <= -3e+81)
		tmp = t_1;
	elseif (x <= -1.25e-195)
		tmp = t_0;
	elseif (x <= -4.5e-279)
		tmp = 4.0;
	elseif (x <= 1e-269)
		tmp = t_0;
	elseif (x <= 2.3e-213)
		tmp = 4.0;
	elseif (x <= 2.7e+44)
		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[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+81], t$95$1, If[LessEqual[x, -1.25e-195], t$95$0, If[LessEqual[x, -4.5e-279], 4.0, If[LessEqual[x, 1e-269], t$95$0, If[LessEqual[x, 2.3e-213], 4.0, If[LessEqual[x, 2.7e+44], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999997e81 or 2.7e44 < 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 77.9%

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

   if -2.99999999999999997e81 < x < -1.25000000000000002e-195 or -4.49999999999999995e-279 < x < 9.9999999999999996e-270 or 2.30000000000000003e-213 < x < 2.7e44

   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 z around inf 62.5%

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

      1. *-commutative 62.5%

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

   6. Simplified 62.5%

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

      1. *-commutative 62.5%

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

      2. clear-num 62.4%

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

      3. un-div-inv 62.4%

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

   8. Applied egg-rr 62.4%

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

      1. associate-/r/ 62.4%

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

   10. Applied egg-rr 62.4%

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

   if -1.25000000000000002e-195 < x < -4.49999999999999995e-279 or 9.9999999999999996e-270 < x < 2.30000000000000003e-213

   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/ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 68.6%

      \[\leadsto 1 + \color{blue}{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+81}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-195}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-279}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 10^{-269}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-213}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 57.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -4 \cdot \frac{z}{y}\\ t_1 := 1 + 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-281}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-212}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -4.0 (/ z y)))) (t_1 (+ 1.0 (* 4.0 (/ x y)))))
   (if (<= x -9e+83)
     t_1
     (if (<= x -4.5e-196)
       t_0
       (if (<= x -2.9e-281)
         4.0
         (if (<= x 1.52e-270)
           t_0
           (if (<= x 2.05e-212) 4.0 (if (<= x 3.1e+46) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (-4.0 * (z / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -9e+83) {
		tmp = t_1;
	} else if (x <= -4.5e-196) {
		tmp = t_0;
	} else if (x <= -2.9e-281) {
		tmp = 4.0;
	} else if (x <= 1.52e-270) {
		tmp = t_0;
	} else if (x <= 2.05e-212) {
		tmp = 4.0;
	} else if (x <= 3.1e+46) {
		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 + ((-4.0d0) * (z / y))
    t_1 = 1.0d0 + (4.0d0 * (x / y))
    if (x <= (-9d+83)) then
        tmp = t_1
    else if (x <= (-4.5d-196)) then
        tmp = t_0
    else if (x <= (-2.9d-281)) then
        tmp = 4.0d0
    else if (x <= 1.52d-270) then
        tmp = t_0
    else if (x <= 2.05d-212) then
        tmp = 4.0d0
    else if (x <= 3.1d+46) 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 + (-4.0 * (z / y));
	double t_1 = 1.0 + (4.0 * (x / y));
	double tmp;
	if (x <= -9e+83) {
		tmp = t_1;
	} else if (x <= -4.5e-196) {
		tmp = t_0;
	} else if (x <= -2.9e-281) {
		tmp = 4.0;
	} else if (x <= 1.52e-270) {
		tmp = t_0;
	} else if (x <= 2.05e-212) {
		tmp = 4.0;
	} else if (x <= 3.1e+46) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (-4.0 * (z / y))
	t_1 = 1.0 + (4.0 * (x / y))
	tmp = 0
	if x <= -9e+83:
		tmp = t_1
	elif x <= -4.5e-196:
		tmp = t_0
	elif x <= -2.9e-281:
		tmp = 4.0
	elif x <= 1.52e-270:
		tmp = t_0
	elif x <= 2.05e-212:
		tmp = 4.0
	elif x <= 3.1e+46:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(-4.0 * Float64(z / y)))
	t_1 = Float64(1.0 + Float64(4.0 * Float64(x / y)))
	tmp = 0.0
	if (x <= -9e+83)
		tmp = t_1;
	elseif (x <= -4.5e-196)
		tmp = t_0;
	elseif (x <= -2.9e-281)
		tmp = 4.0;
	elseif (x <= 1.52e-270)
		tmp = t_0;
	elseif (x <= 2.05e-212)
		tmp = 4.0;
	elseif (x <= 3.1e+46)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (-4.0 * (z / y));
	t_1 = 1.0 + (4.0 * (x / y));
	tmp = 0.0;
	if (x <= -9e+83)
		tmp = t_1;
	elseif (x <= -4.5e-196)
		tmp = t_0;
	elseif (x <= -2.9e-281)
		tmp = 4.0;
	elseif (x <= 1.52e-270)
		tmp = t_0;
	elseif (x <= 2.05e-212)
		tmp = 4.0;
	elseif (x <= 3.1e+46)
		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[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+83], t$95$1, If[LessEqual[x, -4.5e-196], t$95$0, If[LessEqual[x, -2.9e-281], 4.0, If[LessEqual[x, 1.52e-270], t$95$0, If[LessEqual[x, 2.05e-212], 4.0, If[LessEqual[x, 3.1e+46], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.9999999999999999e83 or 3.09999999999999975e46 < 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 77.9%

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

   if -8.9999999999999999e83 < x < -4.5e-196 or -2.8999999999999999e-281 < x < 1.52000000000000004e-270 or 2.05000000000000007e-212 < x < 3.09999999999999975e46

   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 z around inf 62.5%

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

      1. *-commutative 62.5%

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

   6. Simplified 62.5%

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

   if -4.5e-196 < x < -2.8999999999999999e-281 or 1.52000000000000004e-270 < x < 2.05000000000000007e-212

   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/ 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 68.6%

      \[\leadsto 1 + \color{blue}{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+83}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-196}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-281}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{-270}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-212}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+46}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -23 \lor \neg \left(y \leq 2.8 \cdot 10^{+125}\right):\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -23.0) (not (<= y 2.8e+125)))
   (+ 1.0 (+ 3.0 (* x (/ 4.0 y))))
   (+ 1.0 (* 4.0 (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -23.0) || !(y <= 2.8e+125)) {
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	} else {
		tmp = 1.0 + (4.0 * ((x - z) / 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 ((y <= (-23.0d0)) .or. (.not. (y <= 2.8d+125))) then
        tmp = 1.0d0 + (3.0d0 + (x * (4.0d0 / y)))
    else
        tmp = 1.0d0 + (4.0d0 * ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -23.0) || !(y <= 2.8e+125)) {
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	} else {
		tmp = 1.0 + (4.0 * ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -23.0) or not (y <= 2.8e+125):
		tmp = 1.0 + (3.0 + (x * (4.0 / y)))
	else:
		tmp = 1.0 + (4.0 * ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -23.0) || !(y <= 2.8e+125))
		tmp = Float64(1.0 + Float64(3.0 + Float64(x * Float64(4.0 / y))));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -23.0) || ~((y <= 2.8e+125)))
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	else
		tmp = 1.0 + (4.0 * ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -23.0], N[Not[LessEqual[y, 2.8e+125]], $MachinePrecision]], N[(1.0 + N[(3.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -23 or 2.8000000000000001e125 < 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/ 99.8%

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

      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%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 90.8%

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

      1. associate-*r/ 90.8%

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

      2. associate-*l/ 90.7%

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

      3. *-commutative 90.7%

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

   9. Simplified 90.7%

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

   if -23 < y < 2.8000000000000001e125

   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 y around 0 91.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -23 \lor \neg \left(y \leq 2.8 \cdot 10^{+125}\right):\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \end{array} \]

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+78} \lor \neg \left(x \leq 1.15 \cdot 10^{+44}\right):\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + \frac{-4}{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e+78) (not (<= x 1.15e+44)))
   (+ 1.0 (+ 3.0 (* x (/ 4.0 y))))
   (+ 1.0 (+ 3.0 (/ -4.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+78) || !(x <= 1.15e+44)) {
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	} else {
		tmp = 1.0 + (3.0 + (-4.0 / (y / z)));
	}
	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 <= (-1.25d+78)) .or. (.not. (x <= 1.15d+44))) then
        tmp = 1.0d0 + (3.0d0 + (x * (4.0d0 / y)))
    else
        tmp = 1.0d0 + (3.0d0 + ((-4.0d0) / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+78) || !(x <= 1.15e+44)) {
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	} else {
		tmp = 1.0 + (3.0 + (-4.0 / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e+78) or not (x <= 1.15e+44):
		tmp = 1.0 + (3.0 + (x * (4.0 / y)))
	else:
		tmp = 1.0 + (3.0 + (-4.0 / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e+78) || !(x <= 1.15e+44))
		tmp = Float64(1.0 + Float64(3.0 + Float64(x * Float64(4.0 / y))));
	else
		tmp = Float64(1.0 + Float64(3.0 + Float64(-4.0 / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.25e+78) || ~((x <= 1.15e+44)))
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	else
		tmp = 1.0 + (3.0 + (-4.0 / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e+78], N[Not[LessEqual[x, 1.15e+44]], $MachinePrecision]], N[(1.0 + N[(3.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(3.0 + N[(-4.0 / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24999999999999996e78 or 1.15000000000000002e44 < 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 y around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. associate-*l/ 91.5%

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

      3. *-commutative 91.5%

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

   9. Simplified 91.5%

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

   if -1.24999999999999996e78 < x < 1.15000000000000002e44

   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 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 91.7%

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

      1. associate-*r/ 91.7%

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

      2. associate-/l* 91.6%

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

   9. Simplified 91.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+78} \lor \neg \left(x \leq 1.15 \cdot 10^{+44}\right):\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + \frac{-4}{\frac{y}{z}}\right)\\ \end{array} \]

Alternative 6: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+127}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+167}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.6e+127)
   4.0
   (if (<= y 7.6e+167) (+ 1.0 (* 4.0 (/ (- x z) y))) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.6e+127) {
		tmp = 4.0;
	} else if (y <= 7.6e+167) {
		tmp = 1.0 + (4.0 * ((x - z) / 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 <= (-6.6d+127)) then
        tmp = 4.0d0
    else if (y <= 7.6d+167) then
        tmp = 1.0d0 + (4.0d0 * ((x - z) / 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 <= -6.6e+127) {
		tmp = 4.0;
	} else if (y <= 7.6e+167) {
		tmp = 1.0 + (4.0 * ((x - z) / y));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.6e+127:
		tmp = 4.0
	elif y <= 7.6e+167:
		tmp = 1.0 + (4.0 * ((x - z) / y))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.6e+127)
		tmp = 4.0;
	elseif (y <= 7.6e+167)
		tmp = Float64(1.0 + Float64(4.0 * Float64(Float64(x - z) / y)));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.6e+127)
		tmp = 4.0;
	elseif (y <= 7.6e+167)
		tmp = 1.0 + (4.0 * ((x - z) / y));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.6e+127], 4.0, If[LessEqual[y, 7.6e+167], N[(1.0 + N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -6.59999999999999953e127 or 7.59999999999999987e167 < 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/ 99.8%

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

      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 79.5%

      \[\leadsto 1 + \color{blue}{3} \]

   if -6.59999999999999953e127 < y < 7.59999999999999987e167

   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 y around 0 86.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+127}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+167}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 7: 55.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+80} \lor \neg \left(x \leq 8.6 \cdot 10^{+55}\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e+80) (not (<= x 8.6e+55))) (+ 1.0 (* 4.0 (/ x y))) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e+80) || !(x <= 8.6e+55)) {
		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 ((x <= (-3.7d+80)) .or. (.not. (x <= 8.6d+55))) 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 ((x <= -3.7e+80) || !(x <= 8.6e+55)) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e+80) or not (x <= 8.6e+55):
		tmp = 1.0 + (4.0 * (x / y))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e+80) || !(x <= 8.6e+55))
		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 ((x <= -3.7e+80) || ~((x <= 8.6e+55)))
		tmp = 1.0 + (4.0 * (x / y));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e+80], N[Not[LessEqual[x, 8.6e+55]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.69999999999999996e80 or 8.5999999999999998e55 < 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 78.4%

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

   if -3.69999999999999996e80 < x < 8.5999999999999998e55

   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 42.6%

      \[\leadsto 1 + \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+80} \lor \neg \left(x \leq 8.6 \cdot 10^{+55}\right):\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 8: 54.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+81} \lor \neg \left(x \leq 1.7 \cdot 10^{+56}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.05e+81) (not (<= x 1.7e+56))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e+81) || !(x <= 1.7e+56)) {
		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 <= (-1.05d+81)) .or. (.not. (x <= 1.7d+56))) 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 <= -1.05e+81) || !(x <= 1.7e+56)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.05e+81) or not (x <= 1.7e+56):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.05e+81) || !(x <= 1.7e+56))
		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 <= -1.05e+81) || ~((x <= 1.7e+56)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.05e+81], N[Not[LessEqual[x, 1.7e+56]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.0499999999999999e81 or 1.7e56 < 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 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.4%

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

   6. Applied egg-rr 78.4%

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

   7. Taylor expanded in x around inf 76.6%

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

   if -1.0499999999999999e81 < x < 1.7e56

   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 42.6%

      \[\leadsto 1 + \color{blue}{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+81} \lor \neg \left(x \leq 1.7 \cdot 10^{+56}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 9: 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 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 42.7%

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

5. Taylor expanded in y around inf 7.2%

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

6. Final simplification 7.2%

   \[\leadsto 1 \]

Alternative 10: 34.6% 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 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 y around inf 31.4%

   \[\leadsto 1 + \color{blue}{3} \]

5. Final simplification 31.4%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023274 
(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)))