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

Percentage Accurate: 98.0% → 100.0%

Time: 8.6s

Alternatives: 8

Speedup: 1.4×

• 21.2% 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, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z + y, x, 0 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (+ z y) x (- 0.0 z)))

C

double code(double x, double y, double z) {
	return fma((z + y), x, (0.0 - z));
}

Julia

function code(x, y, z)
	return fma(Float64(z + y), x, Float64(0.0 - z))
end

Wolfram

code[x_, y_, z_] := N[(N[(z + y), $MachinePrecision] * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-rgt-identity N/A

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

   5. *-inverses N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. remove-double-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-neg-in N/A

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

   11. *-commutative N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. associate-+r- N/A

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right) - z} \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. sub0-neg N/A

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

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, x, 0 - z\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, x, 0 - z\right) \]

   7. --lowering--.f64 100.0

      \[\leadsto \mathsf{fma}\left(z + y, x, \color{blue}{0 - z}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, x, 0 - z\right)} \]
8. Step-by-step derivation

   1. neg-sub0 N/A

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

   2. neg-lowering-neg.f64 100.0

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(z + y, x, 0 - z\right) \]
11. Add Preprocessing

Alternative 2: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+33}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-30}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.22e+33)
   (* z x)
   (if (<= x -1.22e-85)
     (fma x y 0.0)
     (if (<= x 4.7e-30) (- 0.0 z) (fma x y 0.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e+33) {
		tmp = z * x;
	} else if (x <= -1.22e-85) {
		tmp = fma(x, y, 0.0);
	} else if (x <= 4.7e-30) {
		tmp = 0.0 - z;
	} else {
		tmp = fma(x, y, 0.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.22e+33)
		tmp = Float64(z * x);
	elseif (x <= -1.22e-85)
		tmp = fma(x, y, 0.0);
	elseif (x <= 4.7e-30)
		tmp = Float64(0.0 - z);
	else
		tmp = fma(x, y, 0.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.22e+33], N[(z * x), $MachinePrecision], If[LessEqual[x, -1.22e-85], N[(x * y + 0.0), $MachinePrecision], If[LessEqual[x, 4.7e-30], N[(0.0 - z), $MachinePrecision], N[(x * y + 0.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.22000000000000005e33

   1. Initial program 96.3%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]

   6. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 65.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 65.3

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

   10. Applied egg-rr 65.3%

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

   if -1.22000000000000005e33 < x < -1.22000000000000006e-85 or 4.69999999999999969e-30 < x

   1. Initial program 96.2%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 60.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 60.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   if -1.22000000000000006e-85 < x < 4.69999999999999969e-30

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 77.1

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

   5. Simplified 77.1%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 77.1

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

   7. Applied egg-rr 77.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+33}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-30}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (fma y x (- 0.0 z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = fma(y, x, (0.0 - z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(y * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 95.4%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 99.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-lowering-+.f64 99.0

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

   7. Applied egg-rr 99.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. associate-+r- N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right) - z} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, x, 0 - z\right)} \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{z + y}, x, 0 - z\right) \]

      7. --lowering--.f64 100.0

         \[\leadsto \mathsf{fma}\left(z + y, x, \color{blue}{0 - z}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z + y, x, 0 - z\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. neg-lowering-neg.f64 100.0

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\color{blue}{y}, x, \mathsf{neg}\left(z\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 98.8%

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

4. Final simplification 98.9%

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

Alternative 4: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 310:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= x -7.5e-103) t_0 (if (<= x 310.0) (* z (+ x -1.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (x <= -7.5e-103) {
		tmp = t_0;
	} else if (x <= 310.0) {
		tmp = z * (x + -1.0);
	} 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 + y) * x
    if (x <= (-7.5d-103)) then
        tmp = t_0
    else if (x <= 310.0d0) then
        tmp = z * (x + (-1.0d0))
    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 + y) * x;
	double tmp;
	if (x <= -7.5e-103) {
		tmp = t_0;
	} else if (x <= 310.0) {
		tmp = z * (x + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + y) * x
	tmp = 0
	if x <= -7.5e-103:
		tmp = t_0
	elif x <= 310.0:
		tmp = z * (x + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + y) * x)
	tmp = 0.0
	if (x <= -7.5e-103)
		tmp = t_0;
	elseif (x <= 310.0)
		tmp = Float64(z * Float64(x + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + y) * x;
	tmp = 0.0;
	if (x <= -7.5e-103)
		tmp = t_0;
	elseif (x <= 310.0)
		tmp = z * (x + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -7.5e-103], t$95$0, If[LessEqual[x, 310.0], N[(z * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e-103 or 310 < x

   1. Initial program 96.2%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 93.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

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

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

      4. +-lowering-+.f64 93.0

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

   7. Applied egg-rr 93.0%

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

   if -7.5e-103 < x < 310

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. associate-+r- N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right) - z} \]

   6. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 76.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} - z \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} - z \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(z \cdot x + 0\right) + \left(\mathsf{neg}\left(z\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 76.7

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

   10. Applied egg-rr 76.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-103}:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{elif}\;x \leq 310:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+29}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.95e+153)
   (fma x y 0.0)
   (if (<= y 1.16e+29) (* z (+ x -1.0)) (fma x y 0.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.95e+153) {
		tmp = fma(x, y, 0.0);
	} else if (y <= 1.16e+29) {
		tmp = z * (x + -1.0);
	} else {
		tmp = fma(x, y, 0.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.95e+153)
		tmp = fma(x, y, 0.0);
	elseif (y <= 1.16e+29)
		tmp = Float64(z * Float64(x + -1.0));
	else
		tmp = fma(x, y, 0.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.95e+153], N[(x * y + 0.0), $MachinePrecision], If[LessEqual[y, 1.16e+29], N[(z * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * y + 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9500000000000001e153 or 1.16e29 < y

   1. Initial program 97.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 78.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   if -2.9500000000000001e153 < y < 1.16e29

   1. Initial program 97.6%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-rgt-identity N/A

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

      5. *-inverses N/A

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

      6. associate-/l* N/A

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

      7. associate-*l/ N/A

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

      8. remove-double-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-in N/A

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

      11. *-commutative N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. associate-+r- N/A

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right) - z} \]

   6. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 78.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} - z \]

   8. Simplified 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} - z \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(z \cdot x + 0\right) + \left(\mathsf{neg}\left(z\right)\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. neg-mul-1 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. +-lowering-+.f64 78.1

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

   10. Applied egg-rr 78.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+29}:\\ \;\;\;\;z \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.0) (* z x) (if (<= x 1.0) (- 0.0 z) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = z * x;
	} else if (x <= 1.0) {
		tmp = 0.0 - z;
	} else {
		tmp = z * 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 <= (-1.0d0)) then
        tmp = z * x
    else if (x <= 1.0d0) then
        tmp = 0.0d0 - z
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = z * x;
	} else if (x <= 1.0) {
		tmp = 0.0 - z;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.0:
		tmp = z * x
	elif x <= 1.0:
		tmp = 0.0 - z
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(z * x);
	elseif (x <= 1.0)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = z * x;
	elseif (x <= 1.0)
		tmp = 0.0 - z;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.0], N[(z * x), $MachinePrecision], If[LessEqual[x, 1.0], N[(0.0 - z), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 95.4%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 99.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]

   6. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 50.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]

   8. Simplified 50.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 50.2

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

   10. Applied egg-rr 50.2%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 67.3

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

   5. Simplified 67.3%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 67.3

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

   7. Applied egg-rr 67.3%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-rgt-identity N/A

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

   5. *-inverses N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. remove-double-neg N/A

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

   9. mul-1-neg N/A

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

   10. distribute-lft-neg-in N/A

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

   11. *-commutative N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. associate-+r- N/A

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right) - z} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

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

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

   4. +-lowering-+.f64 100.0

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

7. Applied egg-rr 100.0%

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

Alternative 8: 35.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ 0 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 z))

C

double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - z
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - z;
}

Python

def code(x, y, z):
	return 0.0 - z

Julia

function code(x, y, z)
	return Float64(0.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - z), $MachinePrecision]

Derivation

1. Initial program 97.6%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 34.2

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

5. Simplified 34.2%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 34.2

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

7. Applied egg-rr 34.2%

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

8. Final simplification 34.2%

   \[\leadsto 0 - z \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))