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

Percentage Accurate: 87.8% → 99.8%

Time: 7.5s

Alternatives: 8

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: 87.8% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-291}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-291) t_0 (if (<= t_0 0.0) (* z (- -1.0 (/ x y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-291) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} 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 = (x + y) / (1.0d0 - (y / z))
    if (t_0 <= (-5d-291)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = z * ((-1.0d0) - (x / y))
    else
        tmp = t_0
    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-291) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = z * (-1.0 - (x / y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-291:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = z * (-1.0 - (x / y))
	else:
		tmp = t_0
	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-291)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(z * Float64(-1.0 - Float64(x / y)));
	else
		tmp = t_0;
	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-291)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = z * (-1.0 - (x / y));
	else
		tmp = t_0;
	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[LessEqual[t$95$0, -5e-291], t$95$0, If[LessEqual[t$95$0, 0.0], N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5.0000000000000003e-291 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 -5.0000000000000003e-291 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 6.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(z \cdot \left(x + y\right)\right)\right), \color{blue}{y}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(x + y\right)\right)\right), y\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(0 - \left(x + y\right)\right)\right), y\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(x + y\right)\right)\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(y + x\right)\right)\right), y\right) \]

      11. +-lowering-+.f64 99.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(y, x\right)\right)\right), y\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z + \frac{x \cdot z}{y}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \color{blue}{\left(\frac{x \cdot z}{y}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right)\right) \]

   8. Simplified 99.7%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} - 1\right), \color{blue}{z}\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(1\right)\right)\right), z\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + -1\right), z\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + -1 \cdot \frac{x}{y}\right), z\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right), z\right) \]

       14. unsub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 - \frac{x}{y}\right), z\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \left(\frac{x}{y}\right)\right), z\right) \]

       16. /-lowering-/.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, y\right)\right), z\right) \]

   11. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-291}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{t\_0}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (* z (- -1.0 (/ x y)))))
   (if (<= y -9.5e+59)
     t_1
     (if (<= y -3.4e-17) (/ y t_0) (if (<= y 7e-30) (/ 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 <= -9.5e+59) {
		tmp = t_1;
	} else if (y <= -3.4e-17) {
		tmp = y / t_0;
	} else if (y <= 7e-30) {
		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 <= (-9.5d+59)) then
        tmp = t_1
    else if (y <= (-3.4d-17)) then
        tmp = y / t_0
    else if (y <= 7d-30) 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 <= -9.5e+59) {
		tmp = t_1;
	} else if (y <= -3.4e-17) {
		tmp = y / t_0;
	} else if (y <= 7e-30) {
		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 <= -9.5e+59:
		tmp = t_1
	elif y <= -3.4e-17:
		tmp = y / t_0
	elif y <= 7e-30:
		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 <= -9.5e+59)
		tmp = t_1;
	elseif (y <= -3.4e-17)
		tmp = Float64(y / t_0);
	elseif (y <= 7e-30)
		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 <= -9.5e+59)
		tmp = t_1;
	elseif (y <= -3.4e-17)
		tmp = y / t_0;
	elseif (y <= 7e-30)
		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, -9.5e+59], t$95$1, If[LessEqual[y, -3.4e-17], N[(y / t$95$0), $MachinePrecision], If[LessEqual[y, 7e-30], N[(x / t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.50000000000000023e59 or 7.0000000000000006e-30 < y

   1. Initial program 79.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(z \cdot \left(x + y\right)\right)\right), \color{blue}{y}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(x + y\right)\right)\right), y\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(0 - \left(x + y\right)\right)\right), y\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(x + y\right)\right)\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(y + x\right)\right)\right), y\right) \]

      11. +-lowering-+.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(y, x\right)\right)\right), y\right) \]

   5. Simplified 69.7%

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

   6. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z + \frac{x \cdot z}{y}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \color{blue}{\left(\frac{x \cdot z}{y}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 74.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right)\right) \]

   8. Simplified 74.8%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} - 1\right), \color{blue}{z}\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(1\right)\right)\right), z\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + -1\right), z\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + -1 \cdot \frac{x}{y}\right), z\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right), z\right) \]

       14. unsub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 - \frac{x}{y}\right), z\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \left(\frac{x}{y}\right)\right), z\right) \]

       16. /-lowering-/.f64 76.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, y\right)\right), z\right) \]

   11. Simplified 76.5%

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

   if -9.50000000000000023e59 < y < -3.3999999999999998e-17

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 88.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 88.0%

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

   if -3.3999999999999998e-17 < y < 7.0000000000000006e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 78.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+59}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+27}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-93}:\\ \;\;\;\;\left(0 - z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e+27)
   (+ x y)
   (if (<= z 6.6e-93) (- (- 0.0 z) (/ (* x z) y)) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e+27) {
		tmp = x + y;
	} else if (z <= 6.6e-93) {
		tmp = (0.0 - z) - ((x * z) / 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 (z <= (-6.8d+27)) then
        tmp = x + y
    else if (z <= 6.6d-93) then
        tmp = (0.0d0 - z) - ((x * z) / 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 (z <= -6.8e+27) {
		tmp = x + y;
	} else if (z <= 6.6e-93) {
		tmp = (0.0 - z) - ((x * z) / y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.8e+27:
		tmp = x + y
	elif z <= 6.6e-93:
		tmp = (0.0 - z) - ((x * z) / y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e+27)
		tmp = Float64(x + y);
	elseif (z <= 6.6e-93)
		tmp = Float64(Float64(0.0 - z) - Float64(Float64(x * z) / y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.8e+27)
		tmp = x + y;
	elseif (z <= 6.6e-93)
		tmp = (0.0 - z) - ((x * z) / y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e+27], N[(x + y), $MachinePrecision], If[LessEqual[z, 6.6e-93], N[(N[(0.0 - z), $MachinePrecision] - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8e27 or 6.6000000000000003e-93 < z

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 78.3%

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

   if -6.8e27 < z < 6.6000000000000003e-93

   1. Initial program 81.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(z \cdot \left(x + y\right)\right)\right), \color{blue}{y}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(x + y\right)\right)\right), y\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(0 - \left(x + y\right)\right)\right), y\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(x + y\right)\right)\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(y + x\right)\right)\right), y\right) \]

      11. +-lowering-+.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(y, x\right)\right)\right), y\right) \]

   5. Simplified 72.7%

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

   6. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z + \frac{x \cdot z}{y}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \color{blue}{\left(\frac{x \cdot z}{y}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 75.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right)\right) \]

   8. Simplified 75.3%

      \[\leadsto \color{blue}{0 - \left(z + \frac{x \cdot z}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+27}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-93}:\\ \;\;\;\;\left(0 - z\right) - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.86 \cdot 10^{-28}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (- -1.0 (/ x y)))))
   (if (<= y -1.65e-38) t_0 (if (<= y 1.86e-28) (+ x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -1.65e-38) {
		tmp = t_0;
	} else if (y <= 1.86e-28) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-1.0d0) - (x / y))
    if (y <= (-1.65d-38)) then
        tmp = t_0
    else if (y <= 1.86d-28) then
        tmp = x + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-1.0 - (x / y));
	double tmp;
	if (y <= -1.65e-38) {
		tmp = t_0;
	} else if (y <= 1.86e-28) {
		tmp = x + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-1.0 - (x / y))
	tmp = 0
	if y <= -1.65e-38:
		tmp = t_0
	elif y <= 1.86e-28:
		tmp = x + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -1.65e-38)
		tmp = t_0;
	elseif (y <= 1.86e-28)
		tmp = Float64(x + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-1.0 - (x / y));
	tmp = 0.0;
	if (y <= -1.65e-38)
		tmp = t_0;
	elseif (y <= 1.86e-28)
		tmp = x + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.65e-38], t$95$0, If[LessEqual[y, 1.86e-28], N[(x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6500000000000001e-38 or 1.86e-28 < y

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(z \cdot \left(x + y\right)\right)\right), \color{blue}{y}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \left(x + y\right)\right)\right), y\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(-1 \cdot \left(x + y\right)\right)\right), y\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(\left(x + y\right)\right)\right)\right), y\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(0 - \left(x + y\right)\right)\right), y\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(x + y\right)\right)\right), y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \left(y + x\right)\right)\right), y\right) \]

      11. +-lowering-+.f64 67.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(y, x\right)\right)\right), y\right) \]

   5. Simplified 67.1%

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

   6. Taylor expanded in y around inf

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(z + \frac{x \cdot z}{y}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \color{blue}{\left(\frac{x \cdot z}{y}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(x \cdot z\right), \color{blue}{y}\right)\right)\right) \]

      7. *-lowering-*.f64 71.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, z\right), y\right)\right)\right) \]

   8. Simplified 71.4%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-lft1-in N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} - 1\right), \color{blue}{z}\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + \left(\mathsf{neg}\left(1\right)\right)\right), z\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{x}{y} + -1\right), z\right) \]

       12. +-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + -1 \cdot \frac{x}{y}\right), z\right) \]

       13. mul-1-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\frac{x}{y}\right)\right)\right), z\right) \]

       14. unsub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\left(-1 - \frac{x}{y}\right), z\right) \]

       15. --lowering--.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \left(\frac{x}{y}\right)\right), z\right) \]

       16. /-lowering-/.f64 73.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(x, y\right)\right), z\right) \]

   11. Simplified 73.4%

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

   if -1.6500000000000001e-38 < y < 1.86e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 80.2%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 80.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-38}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1.86 \cdot 10^{-28}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+23}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e+23) (- 0.0 z) (if (<= y 5.5e-28) (+ x y) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e+23) {
		tmp = 0.0 - z;
	} else if (y <= 5.5e-28) {
		tmp = x + y;
	} else {
		tmp = 0.0 - 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 <= (-2d+23)) then
        tmp = 0.0d0 - z
    else if (y <= 5.5d-28) then
        tmp = x + y
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e+23) {
		tmp = 0.0 - z;
	} else if (y <= 5.5e-28) {
		tmp = x + y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e+23:
		tmp = 0.0 - z
	elif y <= 5.5e-28:
		tmp = x + y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e+23)
		tmp = Float64(0.0 - z);
	elseif (y <= 5.5e-28)
		tmp = Float64(x + y);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e+23)
		tmp = 0.0 - z;
	elseif (y <= 5.5e-28)
		tmp = x + y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e+23], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 5.5e-28], N[(x + y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9999999999999998e23 or 5.49999999999999967e-28 < y

   1. Initial program 81.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 56.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 56.4%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 56.4%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 56.4%

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

   if -1.9999999999999998e23 < y < 5.49999999999999967e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(y, \color{blue}{x}\right) \]

   5. Simplified 75.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+23}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-14}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-14) (- 0.0 z) (if (<= y 7.5e-28) x (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-14) {
		tmp = 0.0 - z;
	} else if (y <= 7.5e-28) {
		tmp = x;
	} else {
		tmp = 0.0 - 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 <= (-8.8d-14)) then
        tmp = 0.0d0 - z
    else if (y <= 7.5d-28) then
        tmp = x
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-14) {
		tmp = 0.0 - z;
	} else if (y <= 7.5e-28) {
		tmp = x;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.8e-14:
		tmp = 0.0 - z
	elif y <= 7.5e-28:
		tmp = x
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.8e-14)
		tmp = Float64(0.0 - z);
	elseif (y <= 7.5e-28)
		tmp = x;
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.8e-14)
		tmp = 0.0 - z;
	elseif (y <= 7.5e-28)
		tmp = x;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.8e-14], N[(0.0 - z), $MachinePrecision], If[LessEqual[y, 7.5e-28], x, N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.8000000000000004e-14 or 7.5000000000000003e-28 < y

   1. Initial program 81.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 56.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 56.2%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(z\right) \]

      2. neg-lowering-neg.f64 56.2%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 56.2%

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

   if -8.8000000000000004e-14 < y < 7.5000000000000003e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 61.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-14}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-73}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.7e-179) x (if (<= x 8.5e-73) y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.7e-179) {
		tmp = x;
	} else if (x <= 8.5e-73) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.7d-179)) then
        tmp = x
    else if (x <= 8.5d-73) then
        tmp = y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.7e-179) {
		tmp = x;
	} else if (x <= 8.5e-73) {
		tmp = y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.7e-179:
		tmp = x
	elif x <= 8.5e-73:
		tmp = y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.7e-179)
		tmp = x;
	elseif (x <= 8.5e-73)
		tmp = y;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.7e-179)
		tmp = x;
	elseif (x <= 8.5e-73)
		tmp = y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.7e-179], x, If[LessEqual[x, 8.5e-73], y, x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999988e-179 or 8.4999999999999996e-73 < x

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 42.2%

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

   if -2.69999999999999988e-179 < x < 8.4999999999999996e-73

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(1 - \frac{y}{z}\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      3. /-lowering-/.f64 82.9%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 82.9%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 42.8%

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

Alternative 8: 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 90.3%

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

3. Taylor expanded in y around 0

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

5. Simplified 33.4%

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

Developer Target 1: 93.7% 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 2024158 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

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