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

Percentage Accurate: 87.9% → 99.0%

Time: 6.3s

Alternatives: 7

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: 87.9% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-233} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-233) (not (<= t_0 0.0)))
     t_0
     (- (- z) (/ (* z (+ x z)) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if ((t_0 <= -1e-233) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + 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 = (x + y) / (1.0d0 - (y / z))
    if ((t_0 <= (-1d-233)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = -z - ((z * (x + z)) / 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 <= -1e-233) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = -z - ((z * (x + z)) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if (t_0 <= -1e-233) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = -z - ((z * (x + z)) / 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 <= -1e-233) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(Float64(-z) - Float64(Float64(z * Float64(x + z)) / 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 <= -1e-233) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = -z - ((z * (x + z)) / 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, -1e-233], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[((-z) - N[(N[(z * N[(x + z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -9.99999999999999958e-234 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -9.99999999999999958e-234 < (/.f64 (+.f64 x y) (-.f64 1 (/.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. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. div-sub 100.0%

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

      4. remove-double-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. distribute-lft-out-- 100.0%

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

      8. mul-1-neg 100.0%

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

      9. distribute-neg-frac 100.0%

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

      10. unsub-neg 100.0%

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

      11. mul-1-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. unpow2 100.0%

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

      16. distribute-rgt-out 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-233} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - \frac{z \cdot \left(x + z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-233} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (or (<= t_0 -1e-233) (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 <= -1e-233) || !(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 <= (-1d-233)) .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 <= -1e-233) || !(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 <= -1e-233) 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 <= -1e-233) || !(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 <= -1e-233) || ~((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, -1e-233], 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))) < -9.99999999999999958e-234 or 0.0 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

   1. Initial program 99.8%

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

   if -9.99999999999999958e-234 < (/.f64 (+.f64 x y) (-.f64 1 (/.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 13.2%

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

      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 92.5%

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

      1. mul-1-neg 92.5%

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

      2. associate-/l* 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-/r/ 14.9%

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

      5. distribute-rgt-in 14.9%

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

      6. distribute-neg-in 14.9%

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

      7. *-commutative 14.9%

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

      8. associate-*l/ 79.9%

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

      9. associate-/l* 87.3%

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

      10. *-inverses 87.3%

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

      11. /-rgt-identity 87.3%

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

      12. unsub-neg 87.3%

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

      13. associate-*r/ 100.0%

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

      14. unsub-neg 100.0%

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

      15. mul-1-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. unsub-neg 100.0%

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

      18. mul-1-neg 100.0%

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

      19. associate-/l* 87.3%

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

      20. distribute-neg-frac 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. neg-mul-1 100.0%

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

      7. distribute-rgt-out-- 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -1 \cdot 10^{-233} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \leq 0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.095 \lor \neg \left(y \leq 3 \cdot 10^{+32}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.095) (not (<= y 3e+32))) (* z (- -1.0 (/ x y))) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.095) || !(y <= 3e+32)) {
		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 <= (-0.095d0)) .or. (.not. (y <= 3d+32))) 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 <= -0.095) || !(y <= 3e+32)) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.095) or not (y <= 3e+32):
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.095) || !(y <= 3e+32))
		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 <= -0.095) || ~((y <= 3e+32)))
		tmp = z * (-1.0 - (x / y));
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.095], N[Not[LessEqual[y, 3e+32]], $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 < -0.095000000000000001 or 3e32 < y

   1. Initial program 81.4%

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

      1. clear-num 81.3%

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

      2. associate-/r/ 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in z around 0 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-/l* 80.0%

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

      3. +-commutative 80.0%

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

      4. associate-/r/ 61.9%

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

      5. distribute-rgt-in 61.9%

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

      6. distribute-neg-in 61.9%

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

      7. *-commutative 61.9%

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

      8. associate-*l/ 61.0%

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

      9. associate-/l* 77.4%

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

      10. *-inverses 77.4%

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

      11. /-rgt-identity 77.4%

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

      12. unsub-neg 77.4%

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

      13. associate-*r/ 75.2%

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

      14. unsub-neg 75.2%

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

      15. mul-1-neg 75.2%

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

      16. +-commutative 75.2%

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

      17. unsub-neg 75.2%

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

      18. mul-1-neg 75.2%

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

      19. associate-/l* 77.5%

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

      20. distribute-neg-frac 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. neg-mul-1 75.2%

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

      2. mul-1-neg 75.2%

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

      3. associate-*l/ 80.2%

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

      4. distribute-lft-neg-in 80.2%

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

      5. cancel-sign-sub-inv 80.2%

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

      6. neg-mul-1 80.2%

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

      7. distribute-rgt-out-- 80.1%

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

   10. Simplified 80.1%

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

   if -0.095000000000000001 < y < 3e32

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.5%

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

      1. +-commutative 80.5%

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

   5. Simplified 80.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.095 \lor \neg \left(y \leq 3 \cdot 10^{+32}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+21} \lor \neg \left(y \leq 5.5 \cdot 10^{+61}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.7e+21) (not (<= y 5.5e+61))) (- z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.7e+21) || !(y <= 5.5e+61)) {
		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 <= (-4.7d+21)) .or. (.not. (y <= 5.5d+61))) 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 <= -4.7e+21) || !(y <= 5.5e+61)) {
		tmp = -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.7e+21) or not (y <= 5.5e+61):
		tmp = -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.7e+21) || !(y <= 5.5e+61))
		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 <= -4.7e+21) || ~((y <= 5.5e+61)))
		tmp = -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.7e+21], N[Not[LessEqual[y, 5.5e+61]], $MachinePrecision]], (-z), N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e21 or 5.50000000000000036e61 < y

   1. Initial program 80.4%

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

   3. Taylor expanded in y around inf 64.4%

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

      1. mul-1-neg 64.4%

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

   5. Simplified 64.4%

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

   if -4.7e21 < y < 5.50000000000000036e61

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 78.7%

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

      1. +-commutative 78.7%

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

   5. Simplified 78.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+21} \lor \neg \left(y \leq 5.5 \cdot 10^{+61}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.075 \lor \neg \left(y \leq 3.7 \cdot 10^{+31}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.075) (not (<= y 3.7e+31))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.075) || !(y <= 3.7e+31)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-0.075d0)) .or. (.not. (y <= 3.7d+31))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.075) || !(y <= 3.7e+31)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.075) or not (y <= 3.7e+31):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.075) || !(y <= 3.7e+31))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.075) || ~((y <= 3.7e+31)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.075], N[Not[LessEqual[y, 3.7e+31]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if y < -0.0749999999999999972 or 3.6999999999999998e31 < y

   1. Initial program 81.5%

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

   3. Taylor expanded in y around inf 62.5%

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

      1. mul-1-neg 62.5%

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

   5. Simplified 62.5%

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

   if -0.0749999999999999972 < y < 3.6999999999999998e31

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 60.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.075 \lor \neg \left(y \leq 3.7 \cdot 10^{+31}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-46}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.5e-15) x (if (<= x 2.7e-46) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-15) {
		tmp = x;
	} else if (x <= 2.7e-46) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.5d-15)) then
        tmp = x
    else if (x <= 2.7d-46) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.5e-15) {
		tmp = x;
	} else if (x <= 2.7e-46) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.5e-15:
		tmp = x
	elif x <= 2.7e-46:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.5e-15)
		tmp = x;
	elseif (x <= 2.7e-46)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.5e-15)
		tmp = x;
	elseif (x <= 2.7e-46)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.5e-15], x, If[LessEqual[x, 2.7e-46], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.4999999999999996e-15 or 2.7e-46 < x

   1. Initial program 90.0%

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

   3. Taylor expanded in y around 0 50.3%

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

   if -7.4999999999999996e-15 < x < 2.7e-46

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0 78.4%

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

   4. Taylor expanded in y around 0 39.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-46}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% 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 91.4%

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

3. Taylor expanded in y around 0 35.7%

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

4. Final simplification 35.7%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 94.1% 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 2024018 
(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))))