Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Percentage Accurate: 98.0% → 100.0%

Time: 3.8s

Alternatives: 8

Speedup: 1.3×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y + z, -z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma x (+ y z) (- z)))

C

double code(double x, double y, double z) {
	return fma(x, (y + z), -z);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. sub-neg 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. metadata-eval 99.2%

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

   5. neg-mul-1 99.2%

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

   6. associate-+r+ 99.2%

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

   7. distribute-lft-out 100.0%

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

   8. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x, y + z, -z\right) \]

Alternative 2: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+206}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{+112}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+37}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+206)
   (* x z)
   (if (<= x -7.6e+112)
     (* x y)
     (if (<= x -4.2e+37)
       (* x z)
       (if (<= x -5.3e-69)
         (* x y)
         (if (<= x 1.45e-15) (- z) (if (<= x 2.8e+118) (* x y) (* x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+206) {
		tmp = x * z;
	} else if (x <= -7.6e+112) {
		tmp = x * y;
	} else if (x <= -4.2e+37) {
		tmp = x * z;
	} else if (x <= -5.3e-69) {
		tmp = x * y;
	} else if (x <= 1.45e-15) {
		tmp = -z;
	} else if (x <= 2.8e+118) {
		tmp = x * y;
	} else {
		tmp = x * 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 (x <= (-3.1d+206)) then
        tmp = x * z
    else if (x <= (-7.6d+112)) then
        tmp = x * y
    else if (x <= (-4.2d+37)) then
        tmp = x * z
    else if (x <= (-5.3d-69)) then
        tmp = x * y
    else if (x <= 1.45d-15) then
        tmp = -z
    else if (x <= 2.8d+118) then
        tmp = x * y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+206) {
		tmp = x * z;
	} else if (x <= -7.6e+112) {
		tmp = x * y;
	} else if (x <= -4.2e+37) {
		tmp = x * z;
	} else if (x <= -5.3e-69) {
		tmp = x * y;
	} else if (x <= 1.45e-15) {
		tmp = -z;
	} else if (x <= 2.8e+118) {
		tmp = x * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e+206:
		tmp = x * z
	elif x <= -7.6e+112:
		tmp = x * y
	elif x <= -4.2e+37:
		tmp = x * z
	elif x <= -5.3e-69:
		tmp = x * y
	elif x <= 1.45e-15:
		tmp = -z
	elif x <= 2.8e+118:
		tmp = x * y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+206)
		tmp = Float64(x * z);
	elseif (x <= -7.6e+112)
		tmp = Float64(x * y);
	elseif (x <= -4.2e+37)
		tmp = Float64(x * z);
	elseif (x <= -5.3e-69)
		tmp = Float64(x * y);
	elseif (x <= 1.45e-15)
		tmp = Float64(-z);
	elseif (x <= 2.8e+118)
		tmp = Float64(x * y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.1e+206)
		tmp = x * z;
	elseif (x <= -7.6e+112)
		tmp = x * y;
	elseif (x <= -4.2e+37)
		tmp = x * z;
	elseif (x <= -5.3e-69)
		tmp = x * y;
	elseif (x <= 1.45e-15)
		tmp = -z;
	elseif (x <= 2.8e+118)
		tmp = x * y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e+206], N[(x * z), $MachinePrecision], If[LessEqual[x, -7.6e+112], N[(x * y), $MachinePrecision], If[LessEqual[x, -4.2e+37], N[(x * z), $MachinePrecision], If[LessEqual[x, -5.3e-69], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.45e-15], (-z), If[LessEqual[x, 2.8e+118], N[(x * y), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.09999999999999991e206 or -7.60000000000000015e112 < x < -4.2000000000000002e37 or 2.79999999999999986e118 < x

   1. Initial program 97.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 100.0%

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

      1. +-commutative 100.0%

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

   4. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 88.1%

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

      3. associate-*r/ 83.7%

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

      4. difference-of-squares 87.1%

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

      5. sub-neg 87.1%

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

      6. add-sqr-sqrt 42.5%

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

      7. sqrt-unprod 58.8%

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

      8. sqr-neg 58.8%

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

      9. sqrt-unprod 16.2%

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

      10. add-sqr-sqrt 28.9%

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

      11. pow2 28.9%

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

      12. sub-neg 28.9%

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

      13. add-sqr-sqrt 12.8%

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

      14. sqrt-unprod 42.6%

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

      15. sqr-neg 42.6%

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

      16. sqrt-unprod 44.5%

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

      17. add-sqr-sqrt 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. associate-/l* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-/r* 99.9%

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

      4. *-inverses 99.9%

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

      5. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in z around inf 67.3%

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

   if -3.09999999999999991e206 < x < -7.60000000000000015e112 or -4.2000000000000002e37 < x < -5.3e-69 or 1.45000000000000009e-15 < x < 2.79999999999999986e118

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around inf 62.9%

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

   if -5.3e-69 < x < 1.45000000000000009e-15

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+206}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -7.6 \cdot 10^{+112}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{+37}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+118}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-70} \lor \neg \left(x \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.2e-70) (not (<= x 1.35e-15))) (* x (+ y z)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.2e-70) || !(x <= 1.35e-15)) {
		tmp = x * (y + z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-9.2d-70)) .or. (.not. (x <= 1.35d-15))) then
        tmp = x * (y + z)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.2e-70) || !(x <= 1.35e-15)) {
		tmp = x * (y + z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.2e-70) or not (x <= 1.35e-15):
		tmp = x * (y + z)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.2e-70) || !(x <= 1.35e-15))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.2e-70) || ~((x <= 1.35e-15)))
		tmp = x * (y + z);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.2e-70], N[Not[LessEqual[x, 1.35e-15]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -9.20000000000000002e-70 or 1.35000000000000005e-15 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 93.6%

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

      1. +-commutative 93.6%

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

   4. Simplified 93.6%

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

   if -9.20000000000000002e-70 < x < 1.35000000000000005e-15

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-70} \lor \neg \left(x \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-67} \lor \neg \left(x \leq 1.2 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.3e-67) (not (<= x 1.2e-11))) (* x (+ y z)) (* z (+ x -1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.3e-67) || !(x <= 1.2e-11)) {
		tmp = x * (y + z);
	} else {
		tmp = z * (x + -1.0);
	}
	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) :: tmp
    if ((x <= (-3.3d-67)) .or. (.not. (x <= 1.2d-11))) then
        tmp = x * (y + z)
    else
        tmp = z * (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.3e-67) || !(x <= 1.2e-11)) {
		tmp = x * (y + z);
	} else {
		tmp = z * (x + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.3e-67) or not (x <= 1.2e-11):
		tmp = x * (y + z)
	else:
		tmp = z * (x + -1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.3e-67) || !(x <= 1.2e-11))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(z * Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.3e-67) || ~((x <= 1.2e-11)))
		tmp = x * (y + z);
	else
		tmp = z * (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.3e-67], N[Not[LessEqual[x, 1.2e-11]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3000000000000002e-67 or 1.2000000000000001e-11 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 94.2%

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

      1. +-commutative 94.2%

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

   4. Simplified 94.2%

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

   if -3.3000000000000002e-67 < x < 1.2000000000000001e-11

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around 0 76.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-67} \lor \neg \left(x \leq 1.2 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -54 \lor \neg \left(x \leq 1.15 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -54.0) (not (<= x 1.15e-6))) (* x (+ y z)) (- (* x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -54.0) || !(x <= 1.15e-6)) {
		tmp = x * (y + z);
	} else {
		tmp = (x * 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 ((x <= (-54.0d0)) .or. (.not. (x <= 1.15d-6))) then
        tmp = x * (y + z)
    else
        tmp = (x * y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -54.0) || !(x <= 1.15e-6)) {
		tmp = x * (y + z);
	} else {
		tmp = (x * y) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -54.0) or not (x <= 1.15e-6):
		tmp = x * (y + z)
	else:
		tmp = (x * y) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -54.0) || !(x <= 1.15e-6))
		tmp = Float64(x * Float64(y + z));
	else
		tmp = Float64(Float64(x * y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -54.0) || ~((x <= 1.15e-6)))
		tmp = x * (y + z);
	else
		tmp = (x * y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -54.0], N[Not[LessEqual[x, 1.15e-6]], $MachinePrecision]], N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -54 or 1.15e-6 < x

   1. Initial program 98.5%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

   4. Simplified 98.5%

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

   if -54 < x < 1.15e-6

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. neg-mul-1 100.0%

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

      6. associate-+r+ 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 30.5%

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

      2. flip-+ 14.4%

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

      3. associate-*r/ 14.3%

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

      4. difference-of-squares 14.6%

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

      5. sub-neg 14.6%

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

      6. add-sqr-sqrt 10.4%

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

      7. sqrt-unprod 14.5%

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

      8. sqr-neg 14.5%

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

      9. sqrt-unprod 4.1%

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

      10. add-sqr-sqrt 14.9%

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

      11. pow2 14.9%

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

      12. sub-neg 14.9%

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

      13. add-sqr-sqrt 10.8%

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

      14. sqrt-unprod 13.8%

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

      15. sqr-neg 13.8%

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

      16. sqrt-unprod 4.2%

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

      17. add-sqr-sqrt 14.6%

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

   5. Applied egg-rr 58.0%

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

      1. associate-/l* 14.6%

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

      2. unpow2 14.6%

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

      3. associate-/r* 30.4%

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

      4. *-inverses 30.4%

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

      5. +-commutative 30.4%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 98.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -54 \lor \neg \left(x \leq 1.15 \cdot 10^{-6}\right):\\ \;\;\;\;x \cdot \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - z\\ \end{array} \]

Alternative 6: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.6e-69) (* x y) (if (<= x 1.4e-15) (- z) (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.6e-69) {
		tmp = x * y;
	} else if (x <= 1.4e-15) {
		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 (x <= (-5.6d-69)) then
        tmp = x * y
    else if (x <= 1.4d-15) 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 (x <= -5.6e-69) {
		tmp = x * y;
	} else if (x <= 1.4e-15) {
		tmp = -z;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.6e-69:
		tmp = x * y
	elif x <= 1.4e-15:
		tmp = -z
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.6e-69)
		tmp = Float64(x * y);
	elseif (x <= 1.4e-15)
		tmp = Float64(-z);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.6e-69)
		tmp = x * y;
	elseif (x <= 1.4e-15)
		tmp = -z;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.6e-69], N[(x * y), $MachinePrecision], If[LessEqual[x, 1.4e-15], (-z), N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999959e-69 or 1.40000000000000007e-15 < x

   1. Initial program 98.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in y around inf 50.3%

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

   if -5.59999999999999959e-69 < x < 1.40000000000000007e-15

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]

   2. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-69}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 7: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (+ y z)) z))

C

double code(double x, double y, double z) {
	return (x * (y + z)) - 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 + z)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. sub-neg 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. metadata-eval 99.2%

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

   5. neg-mul-1 99.2%

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

   6. associate-+r+ 99.2%

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

   7. unsub-neg 99.2%

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

   8. +-commutative 99.2%

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

   9. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto x \cdot \left(y + z\right) - z \]

Alternative 8: 36.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.2%

   \[x \cdot y + \left(x - 1\right) \cdot z \]

2. Taylor expanded in x around 0 33.2%

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

   1. mul-1-neg 33.2%

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

4. Simplified 33.2%

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

5. Final simplification 33.2%

   \[\leadsto -z \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))