Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.3% → 99.7%

Time: 5.6s

Alternatives: 10

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (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 + y) / (1.0d0 - (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (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 + y) / (1.0d0 - (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-265} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-265) (not (<= t_0 0.0))) t_0 (- (- z) (* z (/ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-265) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-265)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - (z * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-265) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - (z * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-265) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - (z * (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if ((t_0 <= -5e-265) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) / (1.0 - (y / z));
	tmp = 0.0;
	if ((t_0 <= -5e-265) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - (z * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-265], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -5.0000000000000001e-265 or -0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   if -5.0000000000000001e-265 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -0.0

   1. Initial program 11.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. +-commutative 97.2%

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

      3. *-commutative 97.2%

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

      4. +-commutative 97.2%

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

   4. Simplified 97.2%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. +-commutative 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*r/ 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in x around 0 97.4%

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

      1. associate-*r/ 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-265} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 2: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{y}{t_0}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ y t_0)))
   (if (<= y -1.7e+128)
     (- z)
     (if (<= y -2.5e+58)
       t_1
       (if (<= y 8.5e+22) (/ x t_0) (if (<= y 6.2e+206) t_1 (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -1.7e+128) {
		tmp = -z;
	} else if (y <= -2.5e+58) {
		tmp = t_1;
	} else if (y <= 8.5e+22) {
		tmp = x / t_0;
	} else if (y <= 6.2e+206) {
		tmp = t_1;
	} else {
		tmp = -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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = y / t_0
    if (y <= (-1.7d+128)) then
        tmp = -z
    else if (y <= (-2.5d+58)) then
        tmp = t_1
    else if (y <= 8.5d+22) then
        tmp = x / t_0
    else if (y <= 6.2d+206) then
        tmp = t_1
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = y / t_0;
	double tmp;
	if (y <= -1.7e+128) {
		tmp = -z;
	} else if (y <= -2.5e+58) {
		tmp = t_1;
	} else if (y <= 8.5e+22) {
		tmp = x / t_0;
	} else if (y <= 6.2e+206) {
		tmp = t_1;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = y / t_0
	tmp = 0
	if y <= -1.7e+128:
		tmp = -z
	elif y <= -2.5e+58:
		tmp = t_1
	elif y <= 8.5e+22:
		tmp = x / t_0
	elif y <= 6.2e+206:
		tmp = t_1
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (y <= -1.7e+128)
		tmp = Float64(-z);
	elseif (y <= -2.5e+58)
		tmp = t_1;
	elseif (y <= 8.5e+22)
		tmp = Float64(x / t_0);
	elseif (y <= 6.2e+206)
		tmp = t_1;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = y / t_0;
	tmp = 0.0;
	if (y <= -1.7e+128)
		tmp = -z;
	elseif (y <= -2.5e+58)
		tmp = t_1;
	elseif (y <= 8.5e+22)
		tmp = x / t_0;
	elseif (y <= 6.2e+206)
		tmp = t_1;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, If[LessEqual[y, -1.7e+128], (-z), If[LessEqual[y, -2.5e+58], t$95$1, If[LessEqual[y, 8.5e+22], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 6.2e+206], t$95$1, (-z)]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e128 or 6.19999999999999981e206 < y

   1. Initial program 52.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 80.1%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 80.1%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 80.1%

      \[\leadsto \color{blue}{-z} \]

   if -1.6999999999999999e128 < y < -2.49999999999999993e58 or 8.49999999999999979e22 < y < 6.19999999999999981e206

   1. Initial program 87.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 69.0%

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

   if -2.49999999999999993e58 < y < 8.49999999999999979e22

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 75.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+206}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -0.124:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-40}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) (* z (/ x y)))))
   (if (<= y -0.124)
     t_0
     (if (<= y 7.5e-177)
       (/ x (- 1.0 (/ y z)))
       (if (<= y 6.2e-40) (* (+ x y) (+ 1.0 (/ y z))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -z - (z * (x / y));
	double tmp;
	if (y <= -0.124) {
		tmp = t_0;
	} else if (y <= 7.5e-177) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 6.2e-40) {
		tmp = (x + y) * (1.0 + (y / z));
	} 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) :: tmp
    t_0 = -z - (z * (x / y))
    if (y <= (-0.124d0)) then
        tmp = t_0
    else if (y <= 7.5d-177) then
        tmp = x / (1.0d0 - (y / z))
    else if (y <= 6.2d-40) then
        tmp = (x + y) * (1.0d0 + (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -z - (z * (x / y));
	double tmp;
	if (y <= -0.124) {
		tmp = t_0;
	} else if (y <= 7.5e-177) {
		tmp = x / (1.0 - (y / z));
	} else if (y <= 6.2e-40) {
		tmp = (x + y) * (1.0 + (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -z - (z * (x / y))
	tmp = 0
	if y <= -0.124:
		tmp = t_0
	elif y <= 7.5e-177:
		tmp = x / (1.0 - (y / z))
	elif y <= 6.2e-40:
		tmp = (x + y) * (1.0 + (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - Float64(z * Float64(x / y)))
	tmp = 0.0
	if (y <= -0.124)
		tmp = t_0;
	elseif (y <= 7.5e-177)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	elseif (y <= 6.2e-40)
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -z - (z * (x / y));
	tmp = 0.0;
	if (y <= -0.124)
		tmp = t_0;
	elseif (y <= 7.5e-177)
		tmp = x / (1.0 - (y / z));
	elseif (y <= 6.2e-40)
		tmp = (x + y) * (1.0 + (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.124], t$95$0, If[LessEqual[y, 7.5e-177], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-40], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.124 or 6.20000000000000021e-40 < y

   1. Initial program 75.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. +-commutative 66.7%

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

      3. *-commutative 66.7%

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

      4. +-commutative 66.7%

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

   4. Simplified 66.7%

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

   5. Taylor expanded in y around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in x around 0 71.8%

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

      1. associate-*r/ 78.1%

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

   10. Simplified 78.1%

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

   if -0.124 < y < 7.5e-177

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 84.0%

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

   if 7.5e-177 < y < 6.20000000000000021e-40

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 81.0%

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

      1. associate-/l* 81.0%

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

      2. +-commutative 81.0%

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

      3. associate-/r/ 81.0%

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

      4. +-commutative 81.0%

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

      5. *-lft-identity 81.0%

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

      6. distribute-rgt-in 81.0%

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

      7. +-commutative 81.0%

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

      8. +-commutative 81.0%

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

   4. Simplified 81.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.124:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-40}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 57.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.115:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.115)
   (- z)
   (if (<= y -3.8e-34)
     x
     (if (<= y -1.42e-61)
       (- z)
       (if (<= y -6.5e-112) y (if (<= y 1.76e-37) x (- z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.115) {
		tmp = -z;
	} else if (y <= -3.8e-34) {
		tmp = x;
	} else if (y <= -1.42e-61) {
		tmp = -z;
	} else if (y <= -6.5e-112) {
		tmp = y;
	} else if (y <= 1.76e-37) {
		tmp = x;
	} else {
		tmp = -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 (y <= (-0.115d0)) then
        tmp = -z
    else if (y <= (-3.8d-34)) then
        tmp = x
    else if (y <= (-1.42d-61)) then
        tmp = -z
    else if (y <= (-6.5d-112)) then
        tmp = y
    else if (y <= 1.76d-37) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.115) {
		tmp = -z;
	} else if (y <= -3.8e-34) {
		tmp = x;
	} else if (y <= -1.42e-61) {
		tmp = -z;
	} else if (y <= -6.5e-112) {
		tmp = y;
	} else if (y <= 1.76e-37) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.115:
		tmp = -z
	elif y <= -3.8e-34:
		tmp = x
	elif y <= -1.42e-61:
		tmp = -z
	elif y <= -6.5e-112:
		tmp = y
	elif y <= 1.76e-37:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.115)
		tmp = Float64(-z);
	elseif (y <= -3.8e-34)
		tmp = x;
	elseif (y <= -1.42e-61)
		tmp = Float64(-z);
	elseif (y <= -6.5e-112)
		tmp = y;
	elseif (y <= 1.76e-37)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.115)
		tmp = -z;
	elseif (y <= -3.8e-34)
		tmp = x;
	elseif (y <= -1.42e-61)
		tmp = -z;
	elseif (y <= -6.5e-112)
		tmp = y;
	elseif (y <= 1.76e-37)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.115], (-z), If[LessEqual[y, -3.8e-34], x, If[LessEqual[y, -1.42e-61], (-z), If[LessEqual[y, -6.5e-112], y, If[LessEqual[y, 1.76e-37], x, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.115000000000000005 or -3.8000000000000001e-34 < y < -1.42e-61 or 1.76000000000000006e-37 < y

   1. Initial program 76.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 59.8%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 59.8%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 59.8%

      \[\leadsto \color{blue}{-z} \]

   if -0.115000000000000005 < y < -3.8000000000000001e-34 or -6.49999999999999956e-112 < y < 1.76000000000000006e-37

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around 0 66.3%

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

   if -1.42e-61 < y < -6.49999999999999956e-112

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 46.0%

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

   3. Taylor expanded in y around 0 45.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.115:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.125 \lor \neg \left(y \leq 4.7 \cdot 10^{-39}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.125) (not (<= y 4.7e-39)))
   (- (- z) (* x (/ z y)))
   (/ x (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.125) || !(y <= 4.7e-39)) {
		tmp = -z - (x * (z / y));
	} else {
		tmp = x / (1.0 - (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 ((y <= (-0.125d0)) .or. (.not. (y <= 4.7d-39))) then
        tmp = -z - (x * (z / y))
    else
        tmp = x / (1.0d0 - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.125) || !(y <= 4.7e-39)) {
		tmp = -z - (x * (z / y));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.125) or not (y <= 4.7e-39):
		tmp = -z - (x * (z / y))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.125) || !(y <= 4.7e-39))
		tmp = Float64(Float64(-z) - Float64(x * Float64(z / y)));
	else
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.125) || ~((y <= 4.7e-39)))
		tmp = -z - (x * (z / y));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.125], N[Not[LessEqual[y, 4.7e-39]], $MachinePrecision]], N[((-z) - N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.125 or 4.7000000000000002e-39 < y

   1. Initial program 75.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. +-commutative 66.7%

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

      3. *-commutative 66.7%

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

      4. +-commutative 66.7%

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

   4. Simplified 66.7%

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

   5. Taylor expanded in y around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 74.5%

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

   7. Simplified 74.5%

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

   if -0.125 < y < 4.7000000000000002e-39

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 79.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.125 \lor \neg \left(y \leq 4.7 \cdot 10^{-39}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 6: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 4.3 \cdot 10^{-40}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.082) (not (<= y 4.3e-40)))
   (- (- z) (* z (/ x y)))
   (/ x (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.082) || !(y <= 4.3e-40)) {
		tmp = -z - (z * (x / y));
	} else {
		tmp = x / (1.0 - (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 ((y <= (-0.082d0)) .or. (.not. (y <= 4.3d-40))) then
        tmp = -z - (z * (x / y))
    else
        tmp = x / (1.0d0 - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.082) || !(y <= 4.3e-40)) {
		tmp = -z - (z * (x / y));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.082) or not (y <= 4.3e-40):
		tmp = -z - (z * (x / y))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.082) || !(y <= 4.3e-40))
		tmp = Float64(Float64(-z) - Float64(z * Float64(x / y)));
	else
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.082) || ~((y <= 4.3e-40)))
		tmp = -z - (z * (x / y));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.082], N[Not[LessEqual[y, 4.3e-40]], $MachinePrecision]], N[((-z) - N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0820000000000000034 or 4.3000000000000003e-40 < y

   1. Initial program 75.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around 0 66.7%

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

      1. mul-1-neg 66.7%

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

      2. +-commutative 66.7%

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

      3. *-commutative 66.7%

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

      4. +-commutative 66.7%

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

   4. Simplified 66.7%

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

   5. Taylor expanded in y around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. associate-*r/ 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in x around 0 71.8%

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

      1. associate-*r/ 78.1%

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

   10. Simplified 78.1%

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

   if -0.0820000000000000034 < y < 4.3000000000000003e-40

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 79.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.082 \lor \neg \left(y \leq 4.3 \cdot 10^{-40}\right):\\ \;\;\;\;\left(-z\right) - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 7: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+60}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.8e+60) (- z) (if (<= y 6.2e+25) (/ x (- 1.0 (/ y z))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+60) {
		tmp = -z;
	} else if (y <= 6.2e+25) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -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 (y <= (-3.8d+60)) then
        tmp = -z
    else if (y <= 6.2d+25) then
        tmp = x / (1.0d0 - (y / z))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.8e+60) {
		tmp = -z;
	} else if (y <= 6.2e+25) {
		tmp = x / (1.0 - (y / z));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.8e+60:
		tmp = -z
	elif y <= 6.2e+25:
		tmp = x / (1.0 - (y / z))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.8e+60)
		tmp = Float64(-z);
	elseif (y <= 6.2e+25)
		tmp = Float64(x / Float64(1.0 - Float64(y / z)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.8e+60)
		tmp = -z;
	elseif (y <= 6.2e+25)
		tmp = x / (1.0 - (y / z));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.8e+60], (-z), If[LessEqual[y, 6.2e+25], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -3.80000000000000009e60 or 6.1999999999999996e25 < y

   1. Initial program 70.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 64.3%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 64.3%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 64.3%

      \[\leadsto \color{blue}{-z} \]

   if -3.80000000000000009e60 < y < 6.1999999999999996e25

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around inf 75.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+60}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.15) (- z) (if (<= y 1.76e-37) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.15) {
		tmp = -z;
	} else if (y <= 1.76e-37) {
		tmp = x + y;
	} else {
		tmp = -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 (y <= (-0.15d0)) then
        tmp = -z
    else if (y <= 1.76d-37) then
        tmp = x + y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.15) {
		tmp = -z;
	} else if (y <= 1.76e-37) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.15:
		tmp = -z
	elif y <= 1.76e-37:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.15)
		tmp = Float64(-z);
	elseif (y <= 1.76e-37)
		tmp = Float64(x + y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.15)
		tmp = -z;
	elseif (y <= 1.76e-37)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.15], (-z), If[LessEqual[y, 1.76e-37], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -0.149999999999999994 or 1.76000000000000006e-37 < y

   1. Initial program 75.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around inf 60.4%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 60.4%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 60.4%

      \[\leadsto \color{blue}{-z} \]

   if -0.149999999999999994 < y < 1.76000000000000006e-37

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in z around inf 75.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.76 \cdot 10^{-37}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 40.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.05e-206) x (if (<= x 4.5e-28) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.05e-206) {
		tmp = x;
	} else if (x <= 4.5e-28) {
		tmp = y;
	} 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 (x <= (-2.05d-206)) then
        tmp = x
    else if (x <= 4.5d-28) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.05e-206) {
		tmp = x;
	} else if (x <= 4.5e-28) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.05e-206:
		tmp = x
	elif x <= 4.5e-28:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.05e-206)
		tmp = x;
	elseif (x <= 4.5e-28)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.05e-206)
		tmp = x;
	elseif (x <= 4.5e-28)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.05e-206], x, If[LessEqual[x, 4.5e-28], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000008e-206 or 4.4999999999999998e-28 < x

   1. Initial program 86.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in y around 0 40.8%

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

   if -2.05000000000000008e-206 < x < 4.4999999999999998e-28

   1. Initial program 88.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]

   2. Taylor expanded in x around 0 71.1%

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

   3. Taylor expanded in y around 0 40.3%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-206}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 35.5% accurate, 9.0× 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 87.4%

   \[\frac{x + y}{1 - \frac{y}{z}} \]

2. Taylor expanded in y around 0 33.2%

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

3. Final simplification 33.2%

   \[\leadsto x \]

Developer target: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} 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) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))