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

Percentage Accurate: 88.3% → 99.4%

Time: 8.2s

Alternatives: 12

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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -2e-256)
     t_1
     (if (<= t_1 0.0) (* z (- -1.0 (/ x y))) (* (/ 1.0 t_0) (+ x y))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -2e-256) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (1.0 / t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = (x + y) / t_0
    if (t_1 <= (-2d-256)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = (1.0d0 / t_0) * (x + y)
    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 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -2e-256) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = (1.0 / t_0) * (x + y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = (x + y) / t_0
	tmp = 0
	if t_1 <= -2e-256:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = (1.0 / t_0) * (x + y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(x + y) / t_0)
	tmp = 0.0
	if (t_1 <= -2e-256)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = (x + y) / t_0;
	tmp = 0.0;
	if (t_1 <= -2e-256)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = (1.0 / t_0) * (x + y);
	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[(N[(x + y), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-256], t$95$1, If[LessEqual[t$95$1, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.99999999999999995e-256

   1. Initial program 99.8%

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

   if -1.99999999999999995e-256 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in z around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 6.0%

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

      3. +-commutative 6.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-/l* 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-/l* 99.9%

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

      7. *-lft-identity 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/l* 75.8%

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

      10. associate-/r/ 99.9%

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

      11. distribute-rgt-in 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

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

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

      1. clear-num 99.6%

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

      2. associate-/r/ 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-256}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-256} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -2e-256) (not (<= t_0 0.0))) t_0 (* z (- -1.0 (/ 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 <= -2e-256) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-2d-256)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = z * ((-1.0d0) - (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 <= -2e-256) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -2e-256) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = z * (-1.0 - (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 <= -2e-256) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-1.0 - 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 <= -2e-256) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = z * (-1.0 - (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, -2e-256], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -1.99999999999999995e-256 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -1.99999999999999995e-256 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in z around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. associate-/l* 6.0%

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

      3. +-commutative 6.0%

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

      4. associate-/r/ 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. +-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-/l* 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-/l* 99.9%

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

      7. *-lft-identity 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/l* 75.8%

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

      10. associate-/r/ 99.9%

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

      11. distribute-rgt-in 100.0%

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

      12. distribute-rgt-neg-in 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-256} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 3: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{1 - \frac{y}{z}}\\ t_1 := \frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- 1.0 (/ y z)))) (t_1 (/ 1.0 (+ (/ 1.0 y) (/ -1.0 z)))))
   (if (<= y -1.25e+61)
     t_1
     (if (<= y -1.65e-92)
       t_0
       (if (<= y -3.4e-206)
         (* (+ x y) (+ 1.0 (/ y z)))
         (if (<= y 2.2e-36) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = 1.0 / ((1.0 / y) + (-1.0 / z));
	double tmp;
	if (y <= -1.25e+61) {
		tmp = t_1;
	} else if (y <= -1.65e-92) {
		tmp = t_0;
	} else if (y <= -3.4e-206) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (y <= 2.2e-36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (1.0d0 - (y / z))
    t_1 = 1.0d0 / ((1.0d0 / y) + ((-1.0d0) / z))
    if (y <= (-1.25d+61)) then
        tmp = t_1
    else if (y <= (-1.65d-92)) then
        tmp = t_0
    else if (y <= (-3.4d-206)) then
        tmp = (x + y) * (1.0d0 + (y / z))
    else if (y <= 2.2d-36) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / (1.0 - (y / z));
	double t_1 = 1.0 / ((1.0 / y) + (-1.0 / z));
	double tmp;
	if (y <= -1.25e+61) {
		tmp = t_1;
	} else if (y <= -1.65e-92) {
		tmp = t_0;
	} else if (y <= -3.4e-206) {
		tmp = (x + y) * (1.0 + (y / z));
	} else if (y <= 2.2e-36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (1.0 - (y / z))
	t_1 = 1.0 / ((1.0 / y) + (-1.0 / z))
	tmp = 0
	if y <= -1.25e+61:
		tmp = t_1
	elif y <= -1.65e-92:
		tmp = t_0
	elif y <= -3.4e-206:
		tmp = (x + y) * (1.0 + (y / z))
	elif y <= 2.2e-36:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(1.0 / Float64(Float64(1.0 / y) + Float64(-1.0 / z)))
	tmp = 0.0
	if (y <= -1.25e+61)
		tmp = t_1;
	elseif (y <= -1.65e-92)
		tmp = t_0;
	elseif (y <= -3.4e-206)
		tmp = Float64(Float64(x + y) * Float64(1.0 + Float64(y / z)));
	elseif (y <= 2.2e-36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (1.0 - (y / z));
	t_1 = 1.0 / ((1.0 / y) + (-1.0 / z));
	tmp = 0.0;
	if (y <= -1.25e+61)
		tmp = t_1;
	elseif (y <= -1.65e-92)
		tmp = t_0;
	elseif (y <= -3.4e-206)
		tmp = (x + y) * (1.0 + (y / z));
	elseif (y <= 2.2e-36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(N[(1.0 / y), $MachinePrecision] + N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+61], t$95$1, If[LessEqual[y, -1.65e-92], t$95$0, If[LessEqual[y, -3.4e-206], N[(N[(x + y), $MachinePrecision] * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-36], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25000000000000004e61 or 2.1999999999999999e-36 < y

   1. Initial program 74.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Step-by-step derivation

      1. clear-num 74.4%

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

      2. inv-pow 74.4%

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

   3. Applied egg-rr 74.4%

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

   4. Taylor expanded in z around 0 85.3%

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

      1. +-commutative 85.3%

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

      2. mul-1-neg 85.3%

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

      3. unsub-neg 85.3%

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

      4. +-commutative 85.3%

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

      5. *-commutative 85.3%

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

      6. +-commutative 85.3%

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

   6. Simplified 85.3%

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

   7. Taylor expanded in x around 0 82.3%

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

   if -1.25000000000000004e61 < y < -1.64999999999999999e-92 or -3.39999999999999985e-206 < y < 2.1999999999999999e-36

   1. Initial program 99.0%

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

   2. Taylor expanded in x around inf 83.3%

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

   if -1.64999999999999999e-92 < y < -3.39999999999999985e-206

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 84.3%

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

      1. associate-/l* 84.2%

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

      2. +-commutative 84.2%

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

      3. associate-/r/ 84.3%

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

      4. +-commutative 84.3%

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

      5. *-lft-identity 84.3%

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

      6. distribute-rgt-in 84.3%

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

      7. +-commutative 84.3%

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

      8. +-commutative 84.3%

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

   4. Simplified 84.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-206}:\\ \;\;\;\;\left(x + y\right) \cdot \left(1 + \frac{y}{z}\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \end{array} \]

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 9.7 \cdot 10^{+176} \lor \neg \left(y \leq 7.2 \cdot 10^{+212}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (- -1.0 (/ x y)))))
   (if (<= y -3.5e+63)
     t_1
     (if (<= y 3.2e-16)
       (/ x t_0)
       (if (or (<= y 9.7e+176) (not (<= y 7.2e+212))) t_1 (/ y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -3.5e+63) {
		tmp = t_1;
	} else if (y <= 3.2e-16) {
		tmp = x / t_0;
	} else if ((y <= 9.7e+176) || !(y <= 7.2e+212)) {
		tmp = t_1;
	} else {
		tmp = y / 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 = 1.0d0 - (y / z)
    t_1 = z * ((-1.0d0) - (x / y))
    if (y <= (-3.5d+63)) then
        tmp = t_1
    else if (y <= 3.2d-16) then
        tmp = x / t_0
    else if ((y <= 9.7d+176) .or. (.not. (y <= 7.2d+212))) then
        tmp = t_1
    else
        tmp = y / t_0
    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 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -3.5e+63) {
		tmp = t_1;
	} else if (y <= 3.2e-16) {
		tmp = x / t_0;
	} else if ((y <= 9.7e+176) || !(y <= 7.2e+212)) {
		tmp = t_1;
	} else {
		tmp = y / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -3.5e+63:
		tmp = t_1
	elif y <= 3.2e-16:
		tmp = x / t_0
	elif (y <= 9.7e+176) or not (y <= 7.2e+212):
		tmp = t_1
	else:
		tmp = y / t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -3.5e+63)
		tmp = t_1;
	elseif (y <= 3.2e-16)
		tmp = Float64(x / t_0);
	elseif ((y <= 9.7e+176) || !(y <= 7.2e+212))
		tmp = t_1;
	else
		tmp = Float64(y / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -3.5e+63)
		tmp = t_1;
	elseif (y <= 3.2e-16)
		tmp = x / t_0;
	elseif ((y <= 9.7e+176) || ~((y <= 7.2e+212)))
		tmp = t_1;
	else
		tmp = y / t_0;
	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[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e+63], t$95$1, If[LessEqual[y, 3.2e-16], N[(x / t$95$0), $MachinePrecision], If[Or[LessEqual[y, 9.7e+176], N[Not[LessEqual[y, 7.2e+212]], $MachinePrecision]], t$95$1, N[(y / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000029e63 or 3.20000000000000023e-16 < y < 9.70000000000000041e176 or 7.2e212 < y

   1. Initial program 71.8%

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

   2. Taylor expanded in z around 0 65.2%

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

      1. mul-1-neg 65.2%

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

      2. associate-/l* 50.9%

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

      3. +-commutative 50.9%

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

      4. associate-/r/ 78.9%

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

      5. distribute-rgt-neg-in 78.9%

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

      6. +-commutative 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in y around 0 79.0%

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

   6. Taylor expanded in x around 0 76.0%

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

      1. distribute-lft-out 76.0%

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

      2. associate-/l* 79.0%

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

      3. neg-mul-1 79.0%

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

      4. +-commutative 79.0%

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

      5. +-commutative 79.0%

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

      6. associate-/l* 76.0%

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

      7. *-lft-identity 76.0%

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

      8. *-commutative 76.0%

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

      9. associate-/l* 72.4%

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

      10. associate-/r/ 79.0%

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

      11. distribute-rgt-in 79.0%

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

      12. distribute-rgt-neg-in 79.0%

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

      13. distribute-neg-in 79.0%

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

      14. metadata-eval 79.0%

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

      15. unsub-neg 79.0%

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

   8. Simplified 79.0%

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

   if -3.50000000000000029e63 < y < 3.20000000000000023e-16

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 77.3%

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

   if 9.70000000000000041e176 < y < 7.2e212

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+63}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 9.7 \cdot 10^{+176} \lor \neg \left(y \leq 7.2 \cdot 10^{+212}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 5: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{t_0}\\ \mathbf{elif}\;y \leq 4.05 \cdot 10^{+173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+212}:\\ \;\;\;\;\frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (- (- z) (/ z (/ y x)))))
   (if (<= y -1.25e+61)
     t_1
     (if (<= y 1.35e-16)
       (/ x t_0)
       (if (<= y 4.05e+173)
         t_1
         (if (<= y 5.8e+212) (/ y t_0) (* z (- -1.0 (/ x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.25e+61) {
		tmp = t_1;
	} else if (y <= 1.35e-16) {
		tmp = x / t_0;
	} else if (y <= 4.05e+173) {
		tmp = t_1;
	} else if (y <= 5.8e+212) {
		tmp = y / t_0;
	} else {
		tmp = z * (-1.0 - (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = -z - (z / (y / x))
    if (y <= (-1.25d+61)) then
        tmp = t_1
    else if (y <= 1.35d-16) then
        tmp = x / t_0
    else if (y <= 4.05d+173) then
        tmp = t_1
    else if (y <= 5.8d+212) then
        tmp = y / t_0
    else
        tmp = z * ((-1.0d0) - (x / y))
    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 = -z - (z / (y / x));
	double tmp;
	if (y <= -1.25e+61) {
		tmp = t_1;
	} else if (y <= 1.35e-16) {
		tmp = x / t_0;
	} else if (y <= 4.05e+173) {
		tmp = t_1;
	} else if (y <= 5.8e+212) {
		tmp = y / t_0;
	} else {
		tmp = z * (-1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = -z - (z / (y / x))
	tmp = 0
	if y <= -1.25e+61:
		tmp = t_1
	elif y <= 1.35e-16:
		tmp = x / t_0
	elif y <= 4.05e+173:
		tmp = t_1
	elif y <= 5.8e+212:
		tmp = y / t_0
	else:
		tmp = z * (-1.0 - (x / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	t_1 = Float64(Float64(-z) - Float64(z / Float64(y / x)))
	tmp = 0.0
	if (y <= -1.25e+61)
		tmp = t_1;
	elseif (y <= 1.35e-16)
		tmp = Float64(x / t_0);
	elseif (y <= 4.05e+173)
		tmp = t_1;
	elseif (y <= 5.8e+212)
		tmp = Float64(y / t_0);
	else
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	t_1 = -z - (z / (y / x));
	tmp = 0.0;
	if (y <= -1.25e+61)
		tmp = t_1;
	elseif (y <= 1.35e-16)
		tmp = x / t_0;
	elseif (y <= 4.05e+173)
		tmp = t_1;
	elseif (y <= 5.8e+212)
		tmp = y / t_0;
	else
		tmp = z * (-1.0 - (x / y));
	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[((-z) - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e+61], t$95$1, If[LessEqual[y, 1.35e-16], N[(x / t$95$0), $MachinePrecision], If[LessEqual[y, 4.05e+173], t$95$1, If[LessEqual[y, 5.8e+212], N[(y / t$95$0), $MachinePrecision], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.25000000000000004e61 or 1.35e-16 < y < 4.05000000000000018e173

   1. Initial program 76.2%

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

   2. Taylor expanded in z around 0 66.2%

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

      1. mul-1-neg 66.2%

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

      2. associate-/l* 53.5%

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

      3. +-commutative 53.5%

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

      4. associate-/r/ 77.1%

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

      5. distribute-rgt-neg-in 77.1%

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

      6. +-commutative 77.1%

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

   4. Simplified 77.1%

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

   5. Taylor expanded in y around 0 77.1%

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

      1. distribute-rgt-neg-out 77.1%

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

      2. add-sqr-sqrt 38.2%

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

      3. sqrt-unprod 25.9%

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

      4. sqr-neg 25.9%

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

      5. sqrt-unprod 1.7%

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

      6. add-sqr-sqrt 3.1%

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

      7. *-commutative 3.1%

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

      8. distribute-rgt-in 3.1%

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

      9. *-un-lft-identity 3.1%

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

      10. add-sqr-sqrt 1.7%

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

      11. sqrt-unprod 21.8%

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

      12. sqr-neg 21.8%

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

      13. sqrt-unprod 31.0%

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

      14. add-sqr-sqrt 59.5%

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

      15. *-commutative 59.5%

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

      16. clear-num 59.5%

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

      17. un-div-inv 59.5%

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

      18. add-sqr-sqrt 28.3%

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

      19. sqrt-unprod 59.1%

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

      20. sqr-neg 59.1%

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

      21. sqrt-unprod 38.5%

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

      22. add-sqr-sqrt 77.1%

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

   7. Applied egg-rr 77.1%

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

   if -1.25000000000000004e61 < y < 1.35e-16

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 77.3%

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

   if 4.05000000000000018e173 < y < 5.7999999999999997e212

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   if 5.7999999999999997e212 < y

   1. Initial program 53.2%

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

   2. Taylor expanded in z around 0 61.1%

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

      1. mul-1-neg 61.1%

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

      2. associate-/l* 40.1%

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

      3. +-commutative 40.1%

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

      4. associate-/r/ 86.8%

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

      5. distribute-rgt-neg-in 86.8%

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

      6. +-commutative 86.8%

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

   4. Simplified 86.8%

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

   5. Taylor expanded in y around 0 86.8%

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

   6. Taylor expanded in x around 0 86.8%

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

      1. distribute-lft-out 86.8%

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

      2. associate-/l* 86.8%

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

      3. neg-mul-1 86.8%

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

      4. +-commutative 86.8%

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

      5. +-commutative 86.8%

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

      6. associate-/l* 86.8%

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

      7. *-lft-identity 86.8%

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

      8. *-commutative 86.8%

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

      9. associate-/l* 74.4%

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

      10. associate-/r/ 86.8%

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

      11. distribute-rgt-in 86.8%

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

      12. distribute-rgt-neg-in 86.8%

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

      13. distribute-neg-in 86.8%

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

      14. metadata-eval 86.8%

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

      15. unsub-neg 86.8%

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

   8. Simplified 86.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+61}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 4.05 \cdot 10^{+173}:\\ \;\;\;\;\left(-z\right) - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+212}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-76} \lor \neg \left(y \leq 1.8 \cdot 10^{-25}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-76) (not (<= y 1.8e-25))) (* z (- -1.0 (/ x y))) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-76) || !(y <= 1.8e-25)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = 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 ((y <= (-2.2d-76)) .or. (.not. (y <= 1.8d-25))) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-76) || !(y <= 1.8e-25)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e-76) or not (y <= 1.8e-25):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-76) || !(y <= 1.8e-25))
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e-76) || ~((y <= 1.8e-25)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-76], N[Not[LessEqual[y, 1.8e-25]], $MachinePrecision]], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999999e-76 or 1.8e-25 < y

   1. Initial program 77.9%

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

   2. Taylor expanded in z around 0 62.7%

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

      1. mul-1-neg 62.7%

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

      2. associate-/l* 52.4%

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

      3. +-commutative 52.4%

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

      4. associate-/r/ 73.2%

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

      5. distribute-rgt-neg-in 73.2%

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

      6. +-commutative 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in y around 0 73.2%

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

   6. Taylor expanded in x around 0 71.5%

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

      1. distribute-lft-out 71.5%

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

      2. associate-/l* 73.2%

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

      3. neg-mul-1 73.2%

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

      4. +-commutative 73.2%

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

      5. +-commutative 73.2%

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

      6. associate-/l* 71.5%

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

      7. *-lft-identity 71.5%

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

      8. *-commutative 71.5%

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

      9. associate-/l* 68.8%

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

      10. associate-/r/ 73.2%

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

      11. distribute-rgt-in 73.2%

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

      12. distribute-rgt-neg-in 73.2%

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

      13. distribute-neg-in 73.2%

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

      14. metadata-eval 73.2%

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

      15. unsub-neg 73.2%

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

   8. Simplified 73.2%

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

   if -2.19999999999999999e-76 < y < 1.8e-25

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 80.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-76} \lor \neg \left(y \leq 1.8 \cdot 10^{-25}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 7: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+61} \lor \neg \left(y \leq 2.05 \cdot 10^{-13}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e+61) (not (<= y 2.05e-13)))
   (* z (- -1.0 (/ x y)))
   (/ x (- 1.0 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e+61) || !(y <= 2.05e-13)) {
		tmp = z * (-1.0 - (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 <= (-1.25d+61)) .or. (.not. (y <= 2.05d-13))) then
        tmp = z * ((-1.0d0) - (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 <= -1.25e+61) || !(y <= 2.05e-13)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e+61) or not (y <= 2.05e-13):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e+61) || !(y <= 2.05e-13))
		tmp = Float64(z * Float64(-1.0 - 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 <= -1.25e+61) || ~((y <= 2.05e-13)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e+61], N[Not[LessEqual[y, 2.05e-13]], $MachinePrecision]], N[(z * N[(-1.0 - 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 < -1.25000000000000004e61 or 2.0500000000000001e-13 < y

   1. Initial program 73.9%

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

   2. Taylor expanded in z around 0 63.8%

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

      1. mul-1-neg 63.8%

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

      2. associate-/l* 50.5%

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

      3. +-commutative 50.5%

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

      4. associate-/r/ 76.6%

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

      5. distribute-rgt-neg-in 76.6%

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

      6. +-commutative 76.6%

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

   4. Simplified 76.6%

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

   5. Taylor expanded in y around 0 76.6%

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

   6. Taylor expanded in x around 0 73.8%

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

      1. distribute-lft-out 73.8%

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

      2. associate-/l* 76.6%

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

      3. neg-mul-1 76.6%

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

      4. +-commutative 76.6%

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

      5. +-commutative 76.6%

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

      6. associate-/l* 73.8%

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

      7. *-lft-identity 73.8%

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

      8. *-commutative 73.8%

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

      9. associate-/l* 70.5%

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

      10. associate-/r/ 76.6%

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

      11. distribute-rgt-in 76.6%

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

      12. distribute-rgt-neg-in 76.6%

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

      13. distribute-neg-in 76.6%

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

      14. metadata-eval 76.6%

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

      15. unsub-neg 76.6%

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

   8. Simplified 76.6%

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

   if -1.25000000000000004e61 < y < 2.0500000000000001e-13

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf 77.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+61} \lor \neg \left(y \leq 2.05 \cdot 10^{-13}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]

Alternative 8: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{+61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.62e+61)
   (- z)
   (if (<= y -2.35e-74) (* x (/ (- z) y)) (if (<= y 2.8e-5) (+ x y) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.62e+61) {
		tmp = -z;
	} else if (y <= -2.35e-74) {
		tmp = x * (-z / y);
	} else if (y <= 2.8e-5) {
		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 <= (-1.62d+61)) then
        tmp = -z
    else if (y <= (-2.35d-74)) then
        tmp = x * (-z / y)
    else if (y <= 2.8d-5) 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 <= -1.62e+61) {
		tmp = -z;
	} else if (y <= -2.35e-74) {
		tmp = x * (-z / y);
	} else if (y <= 2.8e-5) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.62e+61:
		tmp = -z
	elif y <= -2.35e-74:
		tmp = x * (-z / y)
	elif y <= 2.8e-5:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.62e+61)
		tmp = Float64(-z);
	elseif (y <= -2.35e-74)
		tmp = Float64(x * Float64(Float64(-z) / y));
	elseif (y <= 2.8e-5)
		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 <= -1.62e+61)
		tmp = -z;
	elseif (y <= -2.35e-74)
		tmp = x * (-z / y);
	elseif (y <= 2.8e-5)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.62e+61], (-z), If[LessEqual[y, -2.35e-74], N[(x * N[((-z) / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-5], N[(x + y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.61999999999999996e61 or 2.79999999999999996e-5 < y

   1. Initial program 73.4%

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

   2. Taylor expanded in y around inf 61.6%

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

      1. mul-1-neg 61.6%

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

   4. Simplified 61.6%

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

   if -1.61999999999999996e61 < y < -2.35000000000000005e-74

   1. Initial program 95.9%

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

   2. Taylor expanded in z around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. associate-/l* 61.7%

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

      3. +-commutative 61.7%

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

      4. distribute-neg-frac 61.7%

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

      5. distribute-neg-in 61.7%

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

      6. +-commutative 61.7%

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

      7. sub-neg 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in y around 0 49.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot x}{y}} \]
   6. Step-by-step derivation

      1. mul-1-neg 49.9%

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

      2. *-commutative 49.9%

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

      3. associate-*r/ 53.2%

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

      4. *-commutative 53.2%

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

      5. distribute-rgt-neg-in 53.2%

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

   7. Simplified 53.2%

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

   if -2.35000000000000005e-74 < y < 2.79999999999999996e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 79.8%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{+61}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.4e+58) (- z) (if (<= y 1.6e-5) (+ x y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.4e+58) {
		tmp = -z;
	} else if (y <= 1.6e-5) {
		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 <= (-1.4d+58)) then
        tmp = -z
    else if (y <= 1.6d-5) 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 <= -1.4e+58) {
		tmp = -z;
	} else if (y <= 1.6e-5) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.4e+58:
		tmp = -z
	elif y <= 1.6e-5:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.4e+58)
		tmp = Float64(-z);
	elseif (y <= 1.6e-5)
		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 <= -1.4e+58)
		tmp = -z;
	elseif (y <= 1.6e-5)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.4e+58], (-z), If[LessEqual[y, 1.6e-5], N[(x + y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999e58 or 1.59999999999999993e-5 < y

   1. Initial program 73.6%

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

   2. Taylor expanded in y around inf 61.2%

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

      1. mul-1-neg 61.2%

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

   4. Simplified 61.2%

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

   if -1.3999999999999999e58 < y < 1.59999999999999993e-5

   1. Initial program 99.2%

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

   2. Taylor expanded in z around inf 72.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 57.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+57) (- z) (if (<= y 1e-11) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+57) {
		tmp = -z;
	} else if (y <= 1e-11) {
		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 <= (-4.8d+57)) then
        tmp = -z
    else if (y <= 1d-11) 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 <= -4.8e+57) {
		tmp = -z;
	} else if (y <= 1e-11) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e+57:
		tmp = -z
	elif y <= 1e-11:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+57)
		tmp = Float64(-z);
	elseif (y <= 1e-11)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e+57)
		tmp = -z;
	elseif (y <= 1e-11)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+57], (-z), If[LessEqual[y, 1e-11], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -4.80000000000000009e57 or 9.99999999999999939e-12 < y

   1. Initial program 74.1%

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

   2. Taylor expanded in y around inf 61.0%

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

      1. mul-1-neg 61.0%

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

   4. Simplified 61.0%

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

   if -4.80000000000000009e57 < y < 9.99999999999999939e-12

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 57.0%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 37.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+85}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.6e+85) y (if (<= y 1.95e-36) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+85) {
		tmp = y;
	} else if (y <= 1.95e-36) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.6d+85)) then
        tmp = y
    else if (y <= 1.95d-36) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.6e+85) {
		tmp = y;
	} else if (y <= 1.95e-36) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.6e+85:
		tmp = y
	elif y <= 1.95e-36:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.6e+85)
		tmp = y;
	elseif (y <= 1.95e-36)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.6e+85)
		tmp = y;
	elseif (y <= 1.95e-36)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.6e+85], y, If[LessEqual[y, 1.95e-36], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5999999999999998e85 or 1.95e-36 < y

   1. Initial program 74.8%

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

   2. Taylor expanded in x around 0 63.0%

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

   3. Taylor expanded in y around 0 23.2%

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

   if -4.5999999999999998e85 < y < 1.95e-36

   1. Initial program 98.5%

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

   2. Taylor expanded in y around 0 56.3%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+85}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 12: 33.9% 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 87.0%

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

2. Taylor expanded in y around 0 31.3%

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

3. Final simplification 31.3%

   \[\leadsto x \]

Developer target: 93.6% 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 2023174 
(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))))