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

Percentage Accurate: 87.6% → 99.6%

Time: 8.2s

Alternatives: 13

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: 87.6% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-272} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(x + z\right)}{-y} - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-272) (not (<= t_0 0.0)))
     t_0
     (- (/ (* z (+ x z)) (- y)) z))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -5e-272) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (x + z)) / -y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-5d-272)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = ((z * (x + z)) / -y) - z
    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-272) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = ((z * (x + z)) / -y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -5e-272) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = ((z * (x + z)) / -y) - z
	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-272) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(z * Float64(x + z)) / Float64(-y)) - z);
	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-272) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = ((z * (x + z)) / -y) - z;
	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-272], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999982e-272 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -4.99999999999999982e-272 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 13.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot z + -1 \cdot \frac{x \cdot z}{y}\right) + \left(-\frac{{z}^{2}}{y}\right)} \]

      2. mul-1-neg 100.0%

         \[\leadsto \left(-1 \cdot z + \color{blue}{\left(-\frac{x \cdot z}{y}\right)}\right) + \left(-\frac{{z}^{2}}{y}\right) \]

      3. unsub-neg 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot z - \frac{x \cdot z}{y}\right)} + \left(-\frac{{z}^{2}}{y}\right) \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{-1 \cdot z - \left(\frac{x \cdot z}{y} - \left(-\frac{{z}^{2}}{y}\right)\right)} \]

      5. distribute-frac-neg 100.0%

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \color{blue}{\frac{-{z}^{2}}{y}}\right) \]

      6. mul-1-neg 100.0%

         \[\leadsto -1 \cdot z - \left(\frac{x \cdot z}{y} - \frac{\color{blue}{-1 \cdot {z}^{2}}}{y}\right) \]

      7. div-sub 100.0%

         \[\leadsto -1 \cdot z - \color{blue}{\frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

      8. unsub-neg 100.0%

         \[\leadsto \color{blue}{-1 \cdot z + \left(-\frac{x \cdot z - -1 \cdot {z}^{2}}{y}\right)} \]

      9. mul-1-neg 100.0%

         \[\leadsto -1 \cdot z + \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y}} \]

      10. +-commutative 100.0%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + -1 \cdot z} \]

      11. mul-1-neg 100.0%

          \[\leadsto -1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(-z\right)} \]

      12. unsub-neg 100.0%

          \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z - -1 \cdot {z}^{2}}{y} - z} \]

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-272} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(x + z\right)}{-y} - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-272} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -5e-272) (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 <= -5e-272) || !(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 <= (-5d-272)) .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 <= -5e-272) || !(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 <= -5e-272) 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 <= -5e-272) || !(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 <= -5e-272) || ~((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, -5e-272], 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 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999982e-272 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   if -4.99999999999999982e-272 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 13.3%

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

      1. clear-num 13.3%

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

      2. associate-/r/ 13.3%

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

   4. Applied egg-rr 13.3%

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

   5. Taylor expanded in z around 0 95.1%

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

      1. associate-*r/ 95.1%

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

      2. +-commutative 95.1%

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

      3. neg-mul-1 95.1%

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

      4. distribute-frac-neg 95.1%

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

      5. distribute-frac-neg2 95.1%

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

      6. sub0-neg 95.1%

         \[\leadsto \frac{z \cdot \left(y + x\right)}{\color{blue}{0 - y}} \]

      7. associate-/l* 99.9%

         \[\leadsto \color{blue}{z \cdot \frac{y + x}{0 - y}} \]

      8. +-commutative 99.9%

         \[\leadsto z \cdot \frac{\color{blue}{x + y}}{0 - y} \]

      9. sub0-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-272} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \frac{1}{t\_0}\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 120000:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (- -1.0 (/ x y)))))
   (if (<= y -4.2e+64)
     t_1
     (if (<= y -5.7e-18)
       (/ y t_0)
       (if (<= y -3.3e-93)
         (* x (/ 1.0 t_0))
         (if (<= y -1.08e-147) (+ x y) (if (<= y 120000.0) (/ x t_0) t_1)))))))

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 <= -4.2e+64) {
		tmp = t_1;
	} else if (y <= -5.7e-18) {
		tmp = y / t_0;
	} else if (y <= -3.3e-93) {
		tmp = x * (1.0 / t_0);
	} else if (y <= -1.08e-147) {
		tmp = x + y;
	} else if (y <= 120000.0) {
		tmp = x / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = z * ((-1.0d0) - (x / y))
    if (y <= (-4.2d+64)) then
        tmp = t_1
    else if (y <= (-5.7d-18)) then
        tmp = y / t_0
    else if (y <= (-3.3d-93)) then
        tmp = x * (1.0d0 / t_0)
    else if (y <= (-1.08d-147)) then
        tmp = x + y
    else if (y <= 120000.0d0) then
        tmp = x / t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -4.2e+64) {
		tmp = t_1;
	} else if (y <= -5.7e-18) {
		tmp = y / t_0;
	} else if (y <= -3.3e-93) {
		tmp = x * (1.0 / t_0);
	} else if (y <= -1.08e-147) {
		tmp = x + y;
	} else if (y <= 120000.0) {
		tmp = x / t_0;
	} else {
		tmp = t_1;
	}
	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 <= -4.2e+64:
		tmp = t_1
	elif y <= -5.7e-18:
		tmp = y / t_0
	elif y <= -3.3e-93:
		tmp = x * (1.0 / t_0)
	elif y <= -1.08e-147:
		tmp = x + y
	elif y <= 120000.0:
		tmp = x / t_0
	else:
		tmp = t_1
	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 <= -4.2e+64)
		tmp = t_1;
	elseif (y <= -5.7e-18)
		tmp = Float64(y / t_0);
	elseif (y <= -3.3e-93)
		tmp = Float64(x * Float64(1.0 / t_0));
	elseif (y <= -1.08e-147)
		tmp = Float64(x + y);
	elseif (y <= 120000.0)
		tmp = Float64(x / t_0);
	else
		tmp = t_1;
	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 <= -4.2e+64)
		tmp = t_1;
	elseif (y <= -5.7e-18)
		tmp = y / t_0;
	elseif (y <= -3.3e-93)
		tmp = x * (1.0 / t_0);
	elseif (y <= -1.08e-147)
		tmp = x + y;
	elseif (y <= 120000.0)
		tmp = x / t_0;
	else
		tmp = t_1;
	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, -4.2e+64], t$95$1, If[LessEqual[y, -5.7e-18], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -3.3e-93], N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.08e-147], N[(x + y), $MachinePrecision], If[LessEqual[y, 120000.0], N[(x / t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.2000000000000001e64 or 1.2e5 < y

   1. Initial program 72.6%

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

      1. clear-num 72.5%

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

      2. associate-/r/ 72.5%

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

   4. Applied egg-rr 72.5%

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

   5. Taylor expanded in z around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. +-commutative 65.7%

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

      3. neg-mul-1 65.7%

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

      4. distribute-frac-neg 65.7%

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

      5. distribute-frac-neg2 65.7%

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

      6. sub0-neg 65.7%

         \[\leadsto \frac{z \cdot \left(y + x\right)}{\color{blue}{0 - y}} \]

      7. associate-/l* 77.9%

         \[\leadsto \color{blue}{z \cdot \frac{y + x}{0 - y}} \]

      8. +-commutative 77.9%

         \[\leadsto z \cdot \frac{\color{blue}{x + y}}{0 - y} \]

      9. sub0-neg 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in x around 0 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. mul-1-neg 77.9%

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

      5. distribute-neg-frac2 77.9%

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

   10. Simplified 77.9%

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

   if -4.2000000000000001e64 < y < -5.69999999999999971e-18

   1. Initial program 99.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

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

   if -5.69999999999999971e-18 < y < -3.3000000000000001e-93

   1. Initial program 99.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. associate-*l/ 69.5%

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

   5. Applied egg-rr 69.5%

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

   if -3.3000000000000001e-93 < y < -1.07999999999999995e-147

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 79.4%

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

      1. +-commutative 79.4%

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

   5. Simplified 79.4%

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

   if -1.07999999999999995e-147 < y < 1.2e5

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 87.0%

      \[\leadsto \color{blue}{\frac{x}{1 - \frac{y}{z}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \frac{1}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 120000:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ t_2 := \frac{x}{t\_0}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-92}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.14 \cdot 10^{-147}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 750:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (- -1.0 (/ x y)))) (t_2 (/ x t_0)))
   (if (<= y -7e+64)
     t_1
     (if (<= y -6.6e-15)
       (/ y t_0)
       (if (<= y -1.8e-92)
         t_2
         (if (<= y -1.14e-147) (+ x y) (if (<= y 750.0) t_2 t_1)))))))

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 t_2 = x / t_0;
	double tmp;
	if (y <= -7e+64) {
		tmp = t_1;
	} else if (y <= -6.6e-15) {
		tmp = y / t_0;
	} else if (y <= -1.8e-92) {
		tmp = t_2;
	} else if (y <= -1.14e-147) {
		tmp = x + y;
	} else if (y <= 750.0) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    t_1 = z * ((-1.0d0) - (x / y))
    t_2 = x / t_0
    if (y <= (-7d+64)) then
        tmp = t_1
    else if (y <= (-6.6d-15)) then
        tmp = y / t_0
    else if (y <= (-1.8d-92)) then
        tmp = t_2
    else if (y <= (-1.14d-147)) then
        tmp = x + y
    else if (y <= 750.0d0) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = z * (-1.0 - (x / y));
	double t_2 = x / t_0;
	double tmp;
	if (y <= -7e+64) {
		tmp = t_1;
	} else if (y <= -6.6e-15) {
		tmp = y / t_0;
	} else if (y <= -1.8e-92) {
		tmp = t_2;
	} else if (y <= -1.14e-147) {
		tmp = x + y;
	} else if (y <= 750.0) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	t_1 = z * (-1.0 - (x / y))
	t_2 = x / t_0
	tmp = 0
	if y <= -7e+64:
		tmp = t_1
	elif y <= -6.6e-15:
		tmp = y / t_0
	elif y <= -1.8e-92:
		tmp = t_2
	elif y <= -1.14e-147:
		tmp = x + y
	elif y <= 750.0:
		tmp = t_2
	else:
		tmp = t_1
	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)))
	t_2 = Float64(x / t_0)
	tmp = 0.0
	if (y <= -7e+64)
		tmp = t_1;
	elseif (y <= -6.6e-15)
		tmp = Float64(y / t_0);
	elseif (y <= -1.8e-92)
		tmp = t_2;
	elseif (y <= -1.14e-147)
		tmp = Float64(x + y);
	elseif (y <= 750.0)
		tmp = t_2;
	else
		tmp = t_1;
	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));
	t_2 = x / t_0;
	tmp = 0.0;
	if (y <= -7e+64)
		tmp = t_1;
	elseif (y <= -6.6e-15)
		tmp = y / t_0;
	elseif (y <= -1.8e-92)
		tmp = t_2;
	elseif (y <= -1.14e-147)
		tmp = x + y;
	elseif (y <= 750.0)
		tmp = t_2;
	else
		tmp = t_1;
	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]}, Block[{t$95$2 = N[(x / t$95$0), $MachinePrecision]}, If[LessEqual[y, -7e+64], t$95$1, If[LessEqual[y, -6.6e-15], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, -1.8e-92], t$95$2, If[LessEqual[y, -1.14e-147], N[(x + y), $MachinePrecision], If[LessEqual[y, 750.0], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.9999999999999997e64 or 750 < y

   1. Initial program 72.6%

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

      1. clear-num 72.5%

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

      2. associate-/r/ 72.5%

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

   4. Applied egg-rr 72.5%

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

   5. Taylor expanded in z around 0 65.7%

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

      1. associate-*r/ 65.7%

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

      2. +-commutative 65.7%

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

      3. neg-mul-1 65.7%

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

      4. distribute-frac-neg 65.7%

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

      5. distribute-frac-neg2 65.7%

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

      6. sub0-neg 65.7%

         \[\leadsto \frac{z \cdot \left(y + x\right)}{\color{blue}{0 - y}} \]

      7. associate-/l* 77.9%

         \[\leadsto \color{blue}{z \cdot \frac{y + x}{0 - y}} \]

      8. +-commutative 77.9%

         \[\leadsto z \cdot \frac{\color{blue}{x + y}}{0 - y} \]

      9. sub0-neg 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in x around 0 77.9%

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

      1. sub-neg 77.9%

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

      2. metadata-eval 77.9%

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

      3. +-commutative 77.9%

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

      4. mul-1-neg 77.9%

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

      5. distribute-neg-frac2 77.9%

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

   10. Simplified 77.9%

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

   if -6.9999999999999997e64 < y < -6.6e-15

   1. Initial program 99.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 72.8%

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

   if -6.6e-15 < y < -1.80000000000000008e-92 or -1.14e-147 < y < 750

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 85.2%

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

   if -1.80000000000000008e-92 < y < -1.14e-147

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 79.4%

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

      1. +-commutative 79.4%

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

   5. Simplified 79.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+64}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq -1.14 \cdot 10^{-147}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq 750:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 16500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.1e+50)
   (- z)
   (if (<= y -4.2e-148)
     (+ x y)
     (if (<= y -1.3e-177)
       (* x (/ z (- y)))
       (if (<= y 16500.0) (+ x y) (* y (/ z (- z y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.1e+50) {
		tmp = -z;
	} else if (y <= -4.2e-148) {
		tmp = x + y;
	} else if (y <= -1.3e-177) {
		tmp = x * (z / -y);
	} else if (y <= 16500.0) {
		tmp = x + y;
	} else {
		tmp = y * (z / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-6.1d+50)) then
        tmp = -z
    else if (y <= (-4.2d-148)) then
        tmp = x + y
    else if (y <= (-1.3d-177)) then
        tmp = x * (z / -y)
    else if (y <= 16500.0d0) then
        tmp = x + y
    else
        tmp = y * (z / (z - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.1e+50) {
		tmp = -z;
	} else if (y <= -4.2e-148) {
		tmp = x + y;
	} else if (y <= -1.3e-177) {
		tmp = x * (z / -y);
	} else if (y <= 16500.0) {
		tmp = x + y;
	} else {
		tmp = y * (z / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.1e+50:
		tmp = -z
	elif y <= -4.2e-148:
		tmp = x + y
	elif y <= -1.3e-177:
		tmp = x * (z / -y)
	elif y <= 16500.0:
		tmp = x + y
	else:
		tmp = y * (z / (z - y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.1e+50)
		tmp = Float64(-z);
	elseif (y <= -4.2e-148)
		tmp = Float64(x + y);
	elseif (y <= -1.3e-177)
		tmp = Float64(x * Float64(z / Float64(-y)));
	elseif (y <= 16500.0)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(z / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.1e+50)
		tmp = -z;
	elseif (y <= -4.2e-148)
		tmp = x + y;
	elseif (y <= -1.3e-177)
		tmp = x * (z / -y);
	elseif (y <= 16500.0)
		tmp = x + y;
	else
		tmp = y * (z / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.1e+50], (-z), If[LessEqual[y, -4.2e-148], N[(x + y), $MachinePrecision], If[LessEqual[y, -1.3e-177], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 16500.0], N[(x + y), $MachinePrecision], N[(y * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.10000000000000026e50

   1. Initial program 65.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.6%

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

      1. mul-1-neg 70.6%

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

   5. Simplified 70.6%

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

   if -6.10000000000000026e50 < y < -4.2e-148 or -1.3e-177 < y < 16500

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 76.6%

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

      1. +-commutative 76.6%

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

   5. Simplified 76.6%

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

   if -4.2e-148 < y < -1.3e-177

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.8%

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

   4. Taylor expanded in y around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. associate-/l* 80.1%

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

      3. distribute-lft-neg-in 80.1%

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

   6. Simplified 80.1%

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

   if 16500 < y

   1. Initial program 80.6%

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

      1. clear-num 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in x around 0 64.4%

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

   6. Taylor expanded in y around inf 80.8%

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

      1. frac-sub 58.9%

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

      2. associate-/r/ 58.9%

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

      3. *-un-lft-identity 58.9%

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

      4. *-rgt-identity 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. associate-*l/ 59.0%

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

      2. *-lft-identity 59.0%

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

      3. associate-/l* 66.7%

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

   10. Simplified 66.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.1 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 16500:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+89}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-5} \lor \neg \left(y \leq 4.8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.8e+89)
   (* z (- -1.0 (/ x y)))
   (if (or (<= y -5e-5) (not (<= y 4.8e+22)))
     (/ 1.0 (+ (/ 1.0 y) (/ -1.0 z)))
     (/ x (- 1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.8e+89) {
		tmp = z * (-1.0 - (x / y));
	} else if ((y <= -5e-5) || !(y <= 4.8e+22)) {
		tmp = 1.0 / ((1.0 / y) + (-1.0 / z));
	} 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 <= (-6.8d+89)) then
        tmp = z * ((-1.0d0) - (x / y))
    else if ((y <= (-5d-5)) .or. (.not. (y <= 4.8d+22))) then
        tmp = 1.0d0 / ((1.0d0 / y) + ((-1.0d0) / z))
    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 <= -6.8e+89) {
		tmp = z * (-1.0 - (x / y));
	} else if ((y <= -5e-5) || !(y <= 4.8e+22)) {
		tmp = 1.0 / ((1.0 / y) + (-1.0 / z));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.8e+89:
		tmp = z * (-1.0 - (x / y))
	elif (y <= -5e-5) or not (y <= 4.8e+22):
		tmp = 1.0 / ((1.0 / y) + (-1.0 / z))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.8e+89)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	elseif ((y <= -5e-5) || !(y <= 4.8e+22))
		tmp = Float64(1.0 / Float64(Float64(1.0 / y) + Float64(-1.0 / z)));
	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 <= -6.8e+89)
		tmp = z * (-1.0 - (x / y));
	elseif ((y <= -5e-5) || ~((y <= 4.8e+22)))
		tmp = 1.0 / ((1.0 / y) + (-1.0 / z));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.8e+89], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -5e-5], N[Not[LessEqual[y, 4.8e+22]], $MachinePrecision]], N[(1.0 / N[(N[(1.0 / y), $MachinePrecision] + N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8000000000000004e89

   1. Initial program 64.0%

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

      1. clear-num 63.9%

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

      2. associate-/r/ 64.0%

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

   4. Applied egg-rr 64.0%

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

   5. Taylor expanded in z around 0 74.5%

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

      1. associate-*r/ 74.5%

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

      2. +-commutative 74.5%

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

      3. neg-mul-1 74.5%

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

      4. distribute-frac-neg 74.5%

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

      5. distribute-frac-neg2 74.5%

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

      6. sub0-neg 74.5%

         \[\leadsto \frac{z \cdot \left(y + x\right)}{\color{blue}{0 - y}} \]

      7. associate-/l* 91.4%

         \[\leadsto \color{blue}{z \cdot \frac{y + x}{0 - y}} \]

      8. +-commutative 91.4%

         \[\leadsto z \cdot \frac{\color{blue}{x + y}}{0 - y} \]

      9. sub0-neg 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in x around 0 91.4%

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

      1. sub-neg 91.4%

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

      2. metadata-eval 91.4%

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

      3. +-commutative 91.4%

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

      4. mul-1-neg 91.4%

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

      5. distribute-neg-frac2 91.4%

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

   10. Simplified 91.4%

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

   if -6.8000000000000004e89 < y < -5.00000000000000024e-5 or 4.8e22 < y

   1. Initial program 82.4%

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

      1. clear-num 82.2%

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

   4. Applied egg-rr 82.2%

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

   5. Taylor expanded in x around 0 66.0%

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

   6. Taylor expanded in y around inf 81.2%

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

   if -5.00000000000000024e-5 < y < 4.8e22

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+89}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-5} \lor \neg \left(y \leq 4.8 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{1}{\frac{1}{y} + \frac{-1}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+51}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+69}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.95e+51)
   (- z)
   (if (<= y -4.2e-148)
     (+ x y)
     (if (<= y -1.3e-177)
       (* x (/ z (- y)))
       (if (<= y 5.5e+69) (+ x y) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.95e+51) {
		tmp = -z;
	} else if (y <= -4.2e-148) {
		tmp = x + y;
	} else if (y <= -1.3e-177) {
		tmp = x * (z / -y);
	} else if (y <= 5.5e+69) {
		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 <= (-2.95d+51)) then
        tmp = -z
    else if (y <= (-4.2d-148)) then
        tmp = x + y
    else if (y <= (-1.3d-177)) then
        tmp = x * (z / -y)
    else if (y <= 5.5d+69) 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 <= -2.95e+51) {
		tmp = -z;
	} else if (y <= -4.2e-148) {
		tmp = x + y;
	} else if (y <= -1.3e-177) {
		tmp = x * (z / -y);
	} else if (y <= 5.5e+69) {
		tmp = x + y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.95e+51:
		tmp = -z
	elif y <= -4.2e-148:
		tmp = x + y
	elif y <= -1.3e-177:
		tmp = x * (z / -y)
	elif y <= 5.5e+69:
		tmp = x + y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.95e+51)
		tmp = Float64(-z);
	elseif (y <= -4.2e-148)
		tmp = Float64(x + y);
	elseif (y <= -1.3e-177)
		tmp = Float64(x * Float64(z / Float64(-y)));
	elseif (y <= 5.5e+69)
		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 <= -2.95e+51)
		tmp = -z;
	elseif (y <= -4.2e-148)
		tmp = x + y;
	elseif (y <= -1.3e-177)
		tmp = x * (z / -y);
	elseif (y <= 5.5e+69)
		tmp = x + y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.95e+51], (-z), If[LessEqual[y, -4.2e-148], N[(x + y), $MachinePrecision], If[LessEqual[y, -1.3e-177], N[(x * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e+69], N[(x + y), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.94999999999999991e51 or 5.50000000000000002e69 < y

   1. Initial program 70.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 67.1%

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

      1. mul-1-neg 67.1%

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

   5. Simplified 67.1%

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

   if -2.94999999999999991e51 < y < -4.2e-148 or -1.3e-177 < y < 5.50000000000000002e69

   1. Initial program 99.2%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 73.9%

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

      1. +-commutative 73.9%

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

   5. Simplified 73.9%

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

   if -4.2e-148 < y < -1.3e-177

   1. Initial program 99.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.8%

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

   4. Taylor expanded in y around inf 80.1%

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

      1. mul-1-neg 80.1%

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

      2. associate-/l* 80.1%

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

      3. distribute-lft-neg-in 80.1%

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

   6. Simplified 80.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+51}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-177}:\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+69}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-6} \lor \neg \left(y \leq 8.5 \cdot 10^{+22}\right):\\ \;\;\;\;y \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+67)
   (- z)
   (if (or (<= y -8.5e-6) (not (<= y 8.5e+22)))
     (* y (/ z (- z y)))
     (/ x (- 1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+67) {
		tmp = -z;
	} else if ((y <= -8.5e-6) || !(y <= 8.5e+22)) {
		tmp = y * (z / (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 <= (-1.8d+67)) then
        tmp = -z
    else if ((y <= (-8.5d-6)) .or. (.not. (y <= 8.5d+22))) then
        tmp = y * (z / (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 <= -1.8e+67) {
		tmp = -z;
	} else if ((y <= -8.5e-6) || !(y <= 8.5e+22)) {
		tmp = y * (z / (z - y));
	} else {
		tmp = x / (1.0 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+67:
		tmp = -z
	elif (y <= -8.5e-6) or not (y <= 8.5e+22):
		tmp = y * (z / (z - y))
	else:
		tmp = x / (1.0 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+67)
		tmp = Float64(-z);
	elseif ((y <= -8.5e-6) || !(y <= 8.5e+22))
		tmp = Float64(y * Float64(z / 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 <= -1.8e+67)
		tmp = -z;
	elseif ((y <= -8.5e-6) || ~((y <= 8.5e+22)))
		tmp = y * (z / (z - y));
	else
		tmp = x / (1.0 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+67], (-z), If[Or[LessEqual[y, -8.5e-6], N[Not[LessEqual[y, 8.5e+22]], $MachinePrecision]], N[(y * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7999999999999999e67

   1. Initial program 63.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.8%

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

      1. mul-1-neg 70.8%

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

   5. Simplified 70.8%

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

   if -1.7999999999999999e67 < y < -8.4999999999999999e-6 or 8.49999999999999979e22 < y

   1. Initial program 83.6%

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

      1. clear-num 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in x around 0 67.5%

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

   6. Taylor expanded in y around inf 81.3%

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

      1. frac-sub 62.8%

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

      2. associate-/r/ 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. *-rgt-identity 62.8%

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

   8. Applied egg-rr 62.8%

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

      1. associate-*l/ 62.9%

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

      2. *-lft-identity 62.9%

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

      3. associate-/l* 69.4%

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

   10. Simplified 69.4%

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

   if -8.4999999999999999e-6 < y < 8.49999999999999979e22

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-6} \lor \neg \left(y \leq 8.5 \cdot 10^{+22}\right):\\ \;\;\;\;y \cdot \frac{z}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+67}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -0.0007:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))))
   (if (<= y -1.8e+67)
     (- z)
     (if (<= y -0.0007)
       (/ y t_0)
       (if (<= y 7.4e+22) (/ x t_0) (* y (/ z (- z y))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double tmp;
	if (y <= -1.8e+67) {
		tmp = -z;
	} else if (y <= -0.0007) {
		tmp = y / t_0;
	} else if (y <= 7.4e+22) {
		tmp = x / t_0;
	} else {
		tmp = y * (z / (z - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y / z)
    if (y <= (-1.8d+67)) then
        tmp = -z
    else if (y <= (-0.0007d0)) then
        tmp = y / t_0
    else if (y <= 7.4d+22) then
        tmp = x / t_0
    else
        tmp = y * (z / (z - 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 tmp;
	if (y <= -1.8e+67) {
		tmp = -z;
	} else if (y <= -0.0007) {
		tmp = y / t_0;
	} else if (y <= 7.4e+22) {
		tmp = x / t_0;
	} else {
		tmp = y * (z / (z - y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (y / z)
	tmp = 0
	if y <= -1.8e+67:
		tmp = -z
	elif y <= -0.0007:
		tmp = y / t_0
	elif y <= 7.4e+22:
		tmp = x / t_0
	else:
		tmp = y * (z / (z - y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(y / z))
	tmp = 0.0
	if (y <= -1.8e+67)
		tmp = Float64(-z);
	elseif (y <= -0.0007)
		tmp = Float64(y / t_0);
	elseif (y <= 7.4e+22)
		tmp = Float64(x / t_0);
	else
		tmp = Float64(y * Float64(z / Float64(z - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (y / z);
	tmp = 0.0;
	if (y <= -1.8e+67)
		tmp = -z;
	elseif (y <= -0.0007)
		tmp = y / t_0;
	elseif (y <= 7.4e+22)
		tmp = x / t_0;
	else
		tmp = y * (z / (z - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+67], (-z), If[LessEqual[y, -0.0007], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 7.4e+22], N[(x / t$95$0), $MachinePrecision], N[(y * N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.7999999999999999e67

   1. Initial program 63.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.8%

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

      1. mul-1-neg 70.8%

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

   5. Simplified 70.8%

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

   if -1.7999999999999999e67 < y < -6.99999999999999993e-4

   1. Initial program 99.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 79.2%

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

   if -6.99999999999999993e-4 < y < 7.3999999999999996e22

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 81.6%

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

   if 7.3999999999999996e22 < y

   1. Initial program 80.0%

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

      1. clear-num 79.8%

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

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in x around 0 64.9%

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

   6. Taylor expanded in y around inf 81.8%

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

      1. frac-sub 59.1%

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

      2. associate-/r/ 59.1%

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

      3. *-un-lft-identity 59.1%

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

      4. *-rgt-identity 59.1%

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

   8. Applied egg-rr 59.1%

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

      1. associate-*l/ 59.3%

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

      2. *-lft-identity 59.3%

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

      3. associate-/l* 67.2%

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

   10. Simplified 67.2%

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

Alternative 10: 56.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+46}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-119}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 25000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.75e+46) (- z) (if (<= y -6e-119) y (if (<= y 25000.0) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.75e+46) {
		tmp = -z;
	} else if (y <= -6e-119) {
		tmp = y;
	} else if (y <= 25000.0) {
		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 <= (-1.75d+46)) then
        tmp = -z
    else if (y <= (-6d-119)) then
        tmp = y
    else if (y <= 25000.0d0) 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 <= -1.75e+46) {
		tmp = -z;
	} else if (y <= -6e-119) {
		tmp = y;
	} else if (y <= 25000.0) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.75e+46:
		tmp = -z
	elif y <= -6e-119:
		tmp = y
	elif y <= 25000.0:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.75e+46)
		tmp = Float64(-z);
	elseif (y <= -6e-119)
		tmp = y;
	elseif (y <= 25000.0)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.75e+46)
		tmp = -z;
	elseif (y <= -6e-119)
		tmp = y;
	elseif (y <= 25000.0)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.75e+46], (-z), If[LessEqual[y, -6e-119], y, If[LessEqual[y, 25000.0], x, (-z)]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.74999999999999992e46 or 25000 < y

   1. Initial program 73.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 63.4%

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

      1. mul-1-neg 63.4%

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

   5. Simplified 63.4%

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

   if -1.74999999999999992e46 < y < -6.0000000000000004e-119

   1. Initial program 99.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.7%

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

   4. Taylor expanded in y around 0 41.2%

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

   if -6.0000000000000004e-119 < y < 25000

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 72.4%

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

Alternative 11: 67.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+52} \lor \neg \left(y \leq 6 \cdot 10^{+69}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.45e+52) (not (<= y 6e+69))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.45e+52) || !(y <= 6e+69)) {
		tmp = -z;
	} 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 <= (-1.45d+52)) .or. (.not. (y <= 6d+69))) then
        tmp = -z
    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 <= -1.45e+52) || !(y <= 6e+69)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.45e+52) or not (y <= 6e+69):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.45e+52) || !(y <= 6e+69))
		tmp = Float64(-z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.45e+52) || ~((y <= 6e+69)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.45e+52], N[Not[LessEqual[y, 6e+69]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45e52 or 5.99999999999999967e69 < y

   1. Initial program 70.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 67.1%

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

      1. mul-1-neg 67.1%

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

   5. Simplified 67.1%

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

   if -1.45e52 < y < 5.99999999999999967e69

   1. Initial program 99.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 71.3%

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

      1. +-commutative 71.3%

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

   5. Simplified 71.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+52} \lor \neg \left(y \leq 6 \cdot 10^{+69}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-118}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.95e-118) y (if (<= y 9e+43) x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.95e-118) {
		tmp = y;
	} else if (y <= 9e+43) {
		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 <= (-1.95d-118)) then
        tmp = y
    else if (y <= 9d+43) 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 <= -1.95e-118) {
		tmp = y;
	} else if (y <= 9e+43) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.95e-118:
		tmp = y
	elif y <= 9e+43:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.95e-118)
		tmp = y;
	elseif (y <= 9e+43)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.95e-118)
		tmp = y;
	elseif (y <= 9e+43)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.95e-118], y, If[LessEqual[y, 9e+43], x, y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.95e-118 or 9e43 < y

   1. Initial program 78.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.8%

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

   4. Taylor expanded in y around 0 22.6%

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

   if -1.95e-118 < y < 9e43

   1. Initial program 99.1%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 69.0%

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

Alternative 13: 34.7% 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. Add Preprocessing

3. Taylor expanded in y around 0 34.4%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer target: 93.2% accurate, 0.5× 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 2024097 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (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))))