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

Percentage Accurate: 99.7% → 99.8%

Time: 7.4s

Alternatives: 9

Speedup: 0.1×

• 20.4% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. associate-+r+ 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. fma-def 99.7%

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

   9. associate-*r* 99.7%

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

   10. associate-*l/ 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-inverses 99.8%

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

   13. metadata-eval 99.8%

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

   14. metadata-eval 99.8%

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

   15. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+118}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+41} \lor \neg \left(y \leq -1.02 \cdot 10^{-5}\right) \land \left(y \leq 2.05 \cdot 10^{+86} \lor \neg \left(y \leq 1.25 \cdot 10^{+140}\right) \land y \leq 9.5 \cdot 10^{+168}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e+118)
   4.0
   (if (or (<= y -5.2e+41)
           (and (not (<= y -1.02e-5))
                (or (<= y 2.05e+86)
                    (and (not (<= y 1.25e+140)) (<= y 9.5e+168)))))
     (* (- x z) (/ 4.0 y))
     4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+118) {
		tmp = 4.0;
	} else if ((y <= -5.2e+41) || (!(y <= -1.02e-5) && ((y <= 2.05e+86) || (!(y <= 1.25e+140) && (y <= 9.5e+168))))) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d+118)) then
        tmp = 4.0d0
    else if ((y <= (-5.2d+41)) .or. (.not. (y <= (-1.02d-5))) .and. (y <= 2.05d+86) .or. (.not. (y <= 1.25d+140)) .and. (y <= 9.5d+168)) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e+118) {
		tmp = 4.0;
	} else if ((y <= -5.2e+41) || (!(y <= -1.02e-5) && ((y <= 2.05e+86) || (!(y <= 1.25e+140) && (y <= 9.5e+168))))) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e+118:
		tmp = 4.0
	elif (y <= -5.2e+41) or (not (y <= -1.02e-5) and ((y <= 2.05e+86) or (not (y <= 1.25e+140) and (y <= 9.5e+168)))):
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e+118)
		tmp = 4.0;
	elseif ((y <= -5.2e+41) || (!(y <= -1.02e-5) && ((y <= 2.05e+86) || (!(y <= 1.25e+140) && (y <= 9.5e+168)))))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e+118)
		tmp = 4.0;
	elseif ((y <= -5.2e+41) || (~((y <= -1.02e-5)) && ((y <= 2.05e+86) || (~((y <= 1.25e+140)) && (y <= 9.5e+168)))))
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e+118], 4.0, If[Or[LessEqual[y, -5.2e+41], And[N[Not[LessEqual[y, -1.02e-5]], $MachinePrecision], Or[LessEqual[y, 2.05e+86], And[N[Not[LessEqual[y, 1.25e+140]], $MachinePrecision], LessEqual[y, 9.5e+168]]]]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999967e117 or -5.2000000000000001e41 < y < -1.0200000000000001e-5 or 2.05e86 < y < 1.25000000000000002e140 or 9.49999999999999979e168 < y

   1. Initial program 98.6%

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

      1. associate-/l* 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 78.9%

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

   if -9.99999999999999967e117 < y < -5.2000000000000001e41 or -1.0200000000000001e-5 < y < 2.05e86 or 1.25000000000000002e140 < y < 9.49999999999999979e168

   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%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 97.8%

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

      1. +-commutative 97.8%

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

      2. distribute-lft-out 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in y around 0 88.4%

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

      1. associate-*r/ 88.4%

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

      2. associate-*l/ 88.1%

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

      3. *-commutative 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+118}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+41} \lor \neg \left(y \leq -1.02 \cdot 10^{-5}\right) \land \left(y \leq 2.05 \cdot 10^{+86} \lor \neg \left(y \leq 1.25 \cdot 10^{+140}\right) \land y \leq 9.5 \cdot 10^{+168}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 3: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+21} \lor \neg \left(z \leq 5.5 \cdot 10^{-36}\right) \land \left(z \leq 45000000000000 \lor \neg \left(z \leq 1.62 \cdot 10^{+148}\right)\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+21)
         (and (not (<= z 5.5e-36))
              (or (<= z 45000000000000.0) (not (<= z 1.62e+148)))))
   (* (- x z) (/ 4.0 y))
   (+ 4.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+21) || (!(z <= 5.5e-36) && ((z <= 45000000000000.0) || !(z <= 1.62e+148)))) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.3d+21)) .or. (.not. (z <= 5.5d-36)) .and. (z <= 45000000000000.0d0) .or. (.not. (z <= 1.62d+148))) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+21) || (!(z <= 5.5e-36) && ((z <= 45000000000000.0) || !(z <= 1.62e+148)))) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e+21) or (not (z <= 5.5e-36) and ((z <= 45000000000000.0) or not (z <= 1.62e+148))):
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+21) || (!(z <= 5.5e-36) && ((z <= 45000000000000.0) || !(z <= 1.62e+148))))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.3e+21) || (~((z <= 5.5e-36)) && ((z <= 45000000000000.0) || ~((z <= 1.62e+148)))))
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+21], And[N[Not[LessEqual[z, 5.5e-36]], $MachinePrecision], Or[LessEqual[z, 45000000000000.0], N[Not[LessEqual[z, 1.62e+148]], $MachinePrecision]]]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e21 or 5.49999999999999984e-36 < z < 4.5e13 or 1.62000000000000006e148 < z

   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%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 96.3%

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

      1. +-commutative 96.3%

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

      2. distribute-lft-out 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in y around 0 89.4%

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

      1. associate-*r/ 89.4%

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

      2. associate-*l/ 89.2%

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

      3. *-commutative 89.2%

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

   9. Simplified 89.2%

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

   if -2.3e21 < z < 5.49999999999999984e-36 or 4.5e13 < z < 1.62000000000000006e148

   1. Initial program 99.3%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 93.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+21} \lor \neg \left(z \leq 5.5 \cdot 10^{-36}\right) \land \left(z \leq 45000000000000 \lor \neg \left(z \leq 1.62 \cdot 10^{+148}\right)\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-37} \lor \neg \left(z \leq 2.15 \cdot 10^{+16}\right) \land z \leq 3.85 \cdot 10^{+133}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-36)
   (+ 1.0 (/ 4.0 (/ y (- x z))))
   (if (or (<= z 2.2e-37) (and (not (<= z 2.15e+16)) (<= z 3.85e+133)))
     (+ 4.0 (* 4.0 (/ x y)))
     (+ 4.0 (* (/ z y) -4.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-36) {
		tmp = 1.0 + (4.0 / (y / (x - z)));
	} else if ((z <= 2.2e-37) || (!(z <= 2.15e+16) && (z <= 3.85e+133))) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + ((z / y) * -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 <= (-2.8d-36)) then
        tmp = 1.0d0 + (4.0d0 / (y / (x - z)))
    else if ((z <= 2.2d-37) .or. (.not. (z <= 2.15d+16)) .and. (z <= 3.85d+133)) then
        tmp = 4.0d0 + (4.0d0 * (x / y))
    else
        tmp = 4.0d0 + ((z / y) * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-36) {
		tmp = 1.0 + (4.0 / (y / (x - z)));
	} else if ((z <= 2.2e-37) || (!(z <= 2.15e+16) && (z <= 3.85e+133))) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + ((z / y) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.8e-36:
		tmp = 1.0 + (4.0 / (y / (x - z)))
	elif (z <= 2.2e-37) or (not (z <= 2.15e+16) and (z <= 3.85e+133)):
		tmp = 4.0 + (4.0 * (x / y))
	else:
		tmp = 4.0 + ((z / y) * -4.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-36)
		tmp = Float64(1.0 + Float64(4.0 / Float64(y / Float64(x - z))));
	elseif ((z <= 2.2e-37) || (!(z <= 2.15e+16) && (z <= 3.85e+133)))
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(4.0 + Float64(Float64(z / y) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.8e-36)
		tmp = 1.0 + (4.0 / (y / (x - z)));
	elseif ((z <= 2.2e-37) || (~((z <= 2.15e+16)) && (z <= 3.85e+133)))
		tmp = 4.0 + (4.0 * (x / y));
	else
		tmp = 4.0 + ((z / y) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-36], N[(1.0 + N[(4.0 / N[(y / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 2.2e-37], And[N[Not[LessEqual[z, 2.15e+16]], $MachinePrecision], LessEqual[z, 3.85e+133]]], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8000000000000001e-36

   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%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 91.9%

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

   if -2.8000000000000001e-36 < z < 2.20000000000000002e-37 or 2.15e16 < z < 3.85e133

   1. Initial program 99.2%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 97.3%

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

   if 2.20000000000000002e-37 < z < 2.15e16 or 3.85e133 < z

   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. associate--l+ 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. associate-+r+ 99.8%

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

      6. *-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. associate-*r* 99.8%

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

      10. associate-*l/ 99.9%

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

      11. associate-/l* 99.9%

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

      12. *-inverses 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 88.5%

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

      1. *-commutative 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-36}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x - z}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-37} \lor \neg \left(z \leq 2.15 \cdot 10^{+16}\right) \land z \leq 3.85 \cdot 10^{+133}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 5: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := \frac{z}{y} \cdot -4\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (* (/ z y) -4.0)))
   (if (<= x -2.7e+62)
     t_0
     (if (<= x 8.8e-282)
       t_1
       (if (<= x 5.5e-236) 4.0 (if (<= x 8e-15) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (z / y) * -4.0;
	double tmp;
	if (x <= -2.7e+62) {
		tmp = t_0;
	} else if (x <= 8.8e-282) {
		tmp = t_1;
	} else if (x <= 5.5e-236) {
		tmp = 4.0;
	} else if (x <= 8e-15) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    t_1 = (z / y) * (-4.0d0)
    if (x <= (-2.7d+62)) then
        tmp = t_0
    else if (x <= 8.8d-282) then
        tmp = t_1
    else if (x <= 5.5d-236) then
        tmp = 4.0d0
    else if (x <= 8d-15) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = (z / y) * -4.0;
	double tmp;
	if (x <= -2.7e+62) {
		tmp = t_0;
	} else if (x <= 8.8e-282) {
		tmp = t_1;
	} else if (x <= 5.5e-236) {
		tmp = 4.0;
	} else if (x <= 8e-15) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = (z / y) * -4.0
	tmp = 0
	if x <= -2.7e+62:
		tmp = t_0
	elif x <= 8.8e-282:
		tmp = t_1
	elif x <= 5.5e-236:
		tmp = 4.0
	elif x <= 8e-15:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(Float64(z / y) * -4.0)
	tmp = 0.0
	if (x <= -2.7e+62)
		tmp = t_0;
	elseif (x <= 8.8e-282)
		tmp = t_1;
	elseif (x <= 5.5e-236)
		tmp = 4.0;
	elseif (x <= 8e-15)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = (z / y) * -4.0;
	tmp = 0.0;
	if (x <= -2.7e+62)
		tmp = t_0;
	elseif (x <= 8.8e-282)
		tmp = t_1;
	elseif (x <= 5.5e-236)
		tmp = 4.0;
	elseif (x <= 8e-15)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[x, -2.7e+62], t$95$0, If[LessEqual[x, 8.8e-282], t$95$1, If[LessEqual[x, 5.5e-236], 4.0, If[LessEqual[x, 8e-15], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e62 or 8.0000000000000006e-15 < x

   1. Initial program 99.1%

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

      1. associate-/l* 99.7%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 96.4%

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

      1. +-commutative 96.4%

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

      2. distribute-lft-out 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in x around inf 73.1%

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

   if -2.7e62 < x < 8.79999999999999925e-282 or 5.49999999999999959e-236 < x < 8.0000000000000006e-15

   1. Initial program 99.9%

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

      1. associate-/l* 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-out 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in z around inf 53.9%

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

      1. *-commutative 53.9%

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

   9. Simplified 53.9%

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

   if 8.79999999999999925e-282 < x < 5.49999999999999959e-236

   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* 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 80.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+62}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-15}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.75 - z\right)}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ 4.0 (/ y (+ x (- (* y 0.75) z))))))

C

double code(double x, double y, double z) {
	return 1.0 + (4.0 / (y / (x + ((y * 0.75) - z))));
}

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 / (y / (x + ((y * 0.75d0) - z))))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + (4.0 / (y / (x + ((y * 0.75) - z))));
}

Python

def code(x, y, z):
	return 1.0 + (4.0 / (y / (x + ((y * 0.75) - z))))

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(4.0 / Float64(y / Float64(x + Float64(Float64(y * 0.75) - z)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + (4.0 / (y / (x + ((y * 0.75) - z))));
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(4.0 / N[(y / N[(x + N[(N[(y * 0.75), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate-/l* 99.7%

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

   2. associate--l+ 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 7: 86.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+62} \lor \neg \left(x \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.9e+62) (not (<= x 1.15e-14)))
   (+ 4.0 (* 4.0 (/ x y)))
   (+ 4.0 (* (/ z y) -4.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e+62) || !(x <= 1.15e-14)) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + ((z / y) * -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 <= (-2.9d+62)) .or. (.not. (x <= 1.15d-14))) then
        tmp = 4.0d0 + (4.0d0 * (x / y))
    else
        tmp = 4.0d0 + ((z / y) * (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.9e+62) || !(x <= 1.15e-14)) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + ((z / y) * -4.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.9e+62) or not (x <= 1.15e-14):
		tmp = 4.0 + (4.0 * (x / y))
	else:
		tmp = 4.0 + ((z / y) * -4.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.9e+62) || !(x <= 1.15e-14))
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(4.0 + Float64(Float64(z / y) * -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.9e+62) || ~((x <= 1.15e-14)))
		tmp = 4.0 + (4.0 * (x / y));
	else
		tmp = 4.0 + ((z / y) * -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.9e+62], N[Not[LessEqual[x, 1.15e-14]], $MachinePrecision]], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999984e62 or 1.14999999999999999e-14 < x

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.7%

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

      11. associate-/l* 99.7%

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

      12. *-inverses 99.7%

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

      13. metadata-eval 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 90.7%

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

   if -2.89999999999999984e62 < x < 1.14999999999999999e-14

   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. associate--l+ 99.8%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.8%

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

      6. *-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. fma-def 99.8%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 90.3%

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

      1. *-commutative 90.3%

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

   6. Simplified 90.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+62} \lor \neg \left(x \leq 1.15 \cdot 10^{-14}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z}{y} \cdot -4\\ \end{array} \]

Alternative 8: 53.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+62} \lor \neg \left(x \leq 1.9 \cdot 10^{+44}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.7e+62) (not (<= x 1.9e+44))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e+62) || !(x <= 1.9e+44)) {
		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 <= (-2.7d+62)) .or. (.not. (x <= 1.9d+44))) 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 <= -2.7e+62) || !(x <= 1.9e+44)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.7e+62) or not (x <= 1.9e+44):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.7e+62) || !(x <= 1.9e+44))
		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 <= -2.7e+62) || ~((x <= 1.9e+44)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.7e+62], N[Not[LessEqual[x, 1.9e+44]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e62 or 1.9000000000000001e44 < x

   1. Initial program 99.0%

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

      1. associate-/l* 99.7%

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

      2. associate--l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 96.1%

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

      1. +-commutative 96.1%

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

      2. distribute-lft-out 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in x around inf 76.4%

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

   if -2.7e62 < x < 1.9000000000000001e44

   1. Initial program 99.9%

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

      1. associate-/l* 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 42.2%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+62} \lor \neg \left(x \leq 1.9 \cdot 10^{+44}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 9: 35.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.6%

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

   1. associate-/l* 99.7%

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

   2. associate--l+ 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in y around inf 31.8%

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

5. Final simplification 31.8%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023238 
(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)))