Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.7% → 96.9%

Time: 3.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-15)
   (fma x (/ y z) x)
   (if (<= z 1e-6) (/ (* x (+ z y)) z) (+ x (* x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-15) {
		tmp = fma(x, (y / z), x);
	} else if (z <= 1e-6) {
		tmp = (x * (z + y)) / z;
	} else {
		tmp = x + (x * (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-15)
		tmp = fma(x, Float64(y / z), x);
	elseif (z <= 1e-6)
		tmp = Float64(Float64(x * Float64(z + y)) / z);
	else
		tmp = Float64(x + Float64(x * Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e-15], N[(x * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1e-6], N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0000000000000001e-15

   1. Initial program 73.5%

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

      1. associate-*r/ 99.9%

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

      2. remove-double-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. *-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-inverses 99.9%

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

      11. distribute-lft-out 99.9%

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

      12. *-inverses 99.9%

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

      13. *-rgt-identity 99.9%

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

      14. fma-def 99.9%

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

   3. Simplified 99.9%

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

   if -1.0000000000000001e-15 < z < 9.99999999999999955e-7

   1. Initial program 95.8%

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

   if 9.99999999999999955e-7 < z

   1. Initial program 70.0%

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

      1. associate-*r/ 99.9%

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

      2. remove-double-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. *-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-inverses 99.9%

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

      11. distribute-lft-out 99.9%

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

      12. *-inverses 99.9%

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

      13. *-rgt-identity 99.9%

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

      14. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-16} \lor \neg \left(z \leq 10^{-6}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9e-16) (not (<= z 1e-6)))
   (+ x (* x (/ y z)))
   (/ (* x (+ z y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-16) || !(z <= 1e-6)) {
		tmp = x + (x * (y / z));
	} else {
		tmp = (x * (z + 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 ((z <= (-9d-16)) .or. (.not. (z <= 1d-6))) then
        tmp = x + (x * (y / z))
    else
        tmp = (x * (z + y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9e-16) || !(z <= 1e-6)) {
		tmp = x + (x * (y / z));
	} else {
		tmp = (x * (z + y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9e-16) or not (z <= 1e-6):
		tmp = x + (x * (y / z))
	else:
		tmp = (x * (z + y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9e-16) || !(z <= 1e-6))
		tmp = Float64(x + Float64(x * Float64(y / z)));
	else
		tmp = Float64(Float64(x * Float64(z + y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9e-16) || ~((z <= 1e-6)))
		tmp = x + (x * (y / z));
	else
		tmp = (x * (z + y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9e-16], N[Not[LessEqual[z, 1e-6]], $MachinePrecision]], N[(x + N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000003e-16 or 9.99999999999999955e-7 < z

   1. Initial program 71.7%

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

      1. associate-*r/ 99.9%

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

      2. remove-double-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. div-sub 99.9%

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

      5. distribute-frac-neg 99.9%

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

      6. *-inverses 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. metadata-eval 99.9%

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

      10. *-inverses 99.9%

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

      11. distribute-lft-out 99.9%

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

      12. *-inverses 99.9%

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

      13. *-rgt-identity 99.9%

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

      14. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -9.0000000000000003e-16 < z < 9.99999999999999955e-7

   1. Initial program 95.8%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-16} \lor \neg \left(z \leq 10^{-6}\right):\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+180}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+79}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.05e+180) x (if (<= z 9.8e+79) (* (+ z y) (/ x z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e+180) {
		tmp = x;
	} else if (z <= 9.8e+79) {
		tmp = (z + y) * (x / z);
	} else {
		tmp = x;
	}
	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 <= (-2.05d+180)) then
        tmp = x
    else if (z <= 9.8d+79) then
        tmp = (z + y) * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e+180) {
		tmp = x;
	} else if (z <= 9.8e+79) {
		tmp = (z + y) * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.05e+180:
		tmp = x
	elif z <= 9.8e+79:
		tmp = (z + y) * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.05e+180)
		tmp = x;
	elseif (z <= 9.8e+79)
		tmp = Float64(Float64(z + y) * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.05e+180)
		tmp = x;
	elseif (z <= 9.8e+79)
		tmp = (z + y) * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.05e+180], x, If[LessEqual[z, 9.8e+79], N[(N[(z + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e180 or 9.7999999999999997e79 < z

   1. Initial program 59.3%

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

      1. associate-*l/ 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in z around inf 88.7%

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

   if -2.05e180 < z < 9.7999999999999997e79

   1. Initial program 92.8%

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

      1. associate-*l/ 91.4%

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

   3. Simplified 91.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+180}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+79}:\\ \;\;\;\;\left(z + y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e-94) x (if (<= z 1.6e-104) (* x (/ y z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-94) {
		tmp = x;
	} else if (z <= 1.6e-104) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	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 <= (-7d-94)) then
        tmp = x
    else if (z <= 1.6d-104) then
        tmp = x * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-94) {
		tmp = x;
	} else if (z <= 1.6e-104) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e-94:
		tmp = x
	elif z <= 1.6e-104:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e-94)
		tmp = x;
	elseif (z <= 1.6e-104)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e-94)
		tmp = x;
	elseif (z <= 1.6e-104)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e-94], x, If[LessEqual[z, 1.6e-104], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999996e-94 or 1.59999999999999994e-104 < z

   1. Initial program 78.2%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 72.4%

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

   if -6.99999999999999996e-94 < z < 1.59999999999999994e-104

   1. Initial program 95.0%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 75.7%

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

   6. Simplified 75.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-105}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4e-108) x (if (<= z 6.3e-105) (* y (/ x z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e-108) {
		tmp = x;
	} else if (z <= 6.3e-105) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	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 <= (-5.4d-108)) then
        tmp = x
    else if (z <= 6.3d-105) then
        tmp = y * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4e-108) {
		tmp = x;
	} else if (z <= 6.3e-105) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.4e-108:
		tmp = x
	elif z <= 6.3e-105:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4e-108)
		tmp = x;
	elseif (z <= 6.3e-105)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.4e-108)
		tmp = x;
	elseif (z <= 6.3e-105)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4e-108], x, If[LessEqual[z, 6.3e-105], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000001e-108 or 6.3e-105 < z

   1. Initial program 78.2%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 72.4%

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

   if -5.4000000000000001e-108 < z < 6.3e-105

   1. Initial program 95.0%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 85.7%

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

      1. associate-/l* 75.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 80.7%

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

   6. Applied egg-rr 80.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{-105}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9e-102) x (if (<= z 4.1e-104) (/ y (/ z x)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-102) {
		tmp = x;
	} else if (z <= 4.1e-104) {
		tmp = y / (z / x);
	} else {
		tmp = x;
	}
	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 <= (-9d-102)) then
        tmp = x
    else if (z <= 4.1d-104) then
        tmp = y / (z / x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9e-102) {
		tmp = x;
	} else if (z <= 4.1e-104) {
		tmp = y / (z / x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9e-102:
		tmp = x
	elif z <= 4.1e-104:
		tmp = y / (z / x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9e-102)
		tmp = x;
	elseif (z <= 4.1e-104)
		tmp = Float64(y / Float64(z / x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9e-102)
		tmp = x;
	elseif (z <= 4.1e-104)
		tmp = y / (z / x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9e-102], x, If[LessEqual[z, 4.1e-104], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999999e-102 or 4.09999999999999984e-104 < z

   1. Initial program 78.2%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 72.4%

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

   if -8.99999999999999999e-102 < z < 4.09999999999999984e-104

   1. Initial program 95.0%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 85.7%

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

      1. associate-*r/ 75.7%

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

   6. Simplified 75.7%

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

      1. associate-*r/ 85.7%

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

      2. *-commutative 85.7%

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

      3. associate-/l* 82.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]

   8. Applied egg-rr 82.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 70.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.8e-94) x (if (<= z 4.1e-104) (/ (* x y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e-94) {
		tmp = x;
	} else if (z <= 4.1e-104) {
		tmp = (x * y) / z;
	} else {
		tmp = x;
	}
	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 <= (-8.8d-94)) then
        tmp = x
    else if (z <= 4.1d-104) then
        tmp = (x * y) / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.8e-94) {
		tmp = x;
	} else if (z <= 4.1e-104) {
		tmp = (x * y) / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.8e-94:
		tmp = x
	elif z <= 4.1e-104:
		tmp = (x * y) / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.8e-94)
		tmp = x;
	elseif (z <= 4.1e-104)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.8e-94)
		tmp = x;
	elseif (z <= 4.1e-104)
		tmp = (x * y) / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.8e-94], x, If[LessEqual[z, 4.1e-104], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -8.80000000000000004e-94 or 4.09999999999999984e-104 < z

   1. Initial program 78.2%

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

      1. associate-*l/ 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 72.4%

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

   if -8.80000000000000004e-94 < z < 4.09999999999999984e-104

   1. Initial program 95.0%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in z around 0 85.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. associate-*r/ 94.4%

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

   2. remove-double-neg 94.4%

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

   3. sub-neg 94.4%

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

   4. div-sub 94.4%

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

   5. distribute-frac-neg 94.4%

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

   6. *-inverses 94.4%

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

   7. metadata-eval 94.4%

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

   8. sub-neg 94.4%

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

   9. metadata-eval 94.4%

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

   10. *-inverses 94.4%

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

   11. distribute-lft-out 94.4%

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

   12. *-inverses 94.4%

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

   13. *-rgt-identity 94.4%

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

   14. fma-def 94.4%

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

3. Simplified 94.4%

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

   1. fma-udef 94.4%

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

5. Applied egg-rr 94.4%

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

6. Final simplification 94.4%

   \[\leadsto x + x \cdot \frac{y}{z} \]

Alternative 9: 50.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 84.4%

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

   1. associate-*l/ 83.6%

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

3. Simplified 83.6%

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

4. Taylor expanded in z around inf 50.4%

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

5. Final simplification 50.4%

   \[\leadsto x \]

Developer target: 95.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z (+ y z))))

C

double code(double x, double y, double z) {
	return x / (z / (y + 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 = x / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023308 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))