Expression 1, p15

Average Error: 0.4 → 0.0

Time: 3.3s

Precision: binary64

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]

Math

\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]

↓

\[\log \left(\left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot e^{b}\right) \cdot e^{a}\right) \]

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

↓

(FPCore (a b c d e)
 :precision binary64
 (log (* (* (* (* (exp e) (exp d)) (exp c)) (exp b)) (exp a))))

C

double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}

↓

double code(double a, double b, double c, double d, double e) {
	return log(((((exp(e) * exp(d)) * exp(c)) * exp(b)) * exp(a)));
}

Fortran

real(8) function code(a, b, c, d, e)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8), intent (in) :: e
    code = (((e + d) + c) + b) + a
end function

↓

real(8) function code(a, b, c, d, e)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8), intent (in) :: e
    code = log(((((exp(e) * exp(d)) * exp(c)) * exp(b)) * exp(a)))
end function

Java

public static double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}

↓

public static double code(double a, double b, double c, double d, double e) {
	return Math.log(((((Math.exp(e) * Math.exp(d)) * Math.exp(c)) * Math.exp(b)) * Math.exp(a)));
}

Python

def code(a, b, c, d, e):
	return (((e + d) + c) + b) + a

↓

def code(a, b, c, d, e):
	return math.log(((((math.exp(e) * math.exp(d)) * math.exp(c)) * math.exp(b)) * math.exp(a)))

Julia

function code(a, b, c, d, e)
	return Float64(Float64(Float64(Float64(e + d) + c) + b) + a)
end

↓

function code(a, b, c, d, e)
	return log(Float64(Float64(Float64(Float64(exp(e) * exp(d)) * exp(c)) * exp(b)) * exp(a)))
end

MATLAB

function tmp = code(a, b, c, d, e)
	tmp = (((e + d) + c) + b) + a;
end

↓

function tmp = code(a, b, c, d, e)
	tmp = log(((((exp(e) * exp(d)) * exp(c)) * exp(b)) * exp(a)));
end

Wolfram

code[a_, b_, c_, d_, e_] := N[(N[(N[(N[(e + d), $MachinePrecision] + c), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision]

↓

code[a_, b_, c_, d_, e_] := N[Log[N[(N[(N[(N[(N[Exp[e], $MachinePrecision] * N[Exp[d], $MachinePrecision]), $MachinePrecision] * N[Exp[c], $MachinePrecision]), $MachinePrecision] * N[Exp[b], $MachinePrecision]), $MachinePrecision] * N[Exp[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original	0.4
Target	0.2
Herbie	0.0

\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

Derivation

1. Initial program 0.4

   \[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]

2. Applied add-log-exp_binary64 0.4

   \[\leadsto \left(\left(\left(e + d\right) + c\right) + b\right) + \color{blue}{\log \left(e^{a}\right)} \]

3. Applied add-log-exp_binary64 0.4

   \[\leadsto \left(\left(\left(e + d\right) + c\right) + \color{blue}{\log \left(e^{b}\right)}\right) + \log \left(e^{a}\right) \]

4. Applied add-log-exp_binary64 0.4

   \[\leadsto \left(\left(\left(e + d\right) + \color{blue}{\log \left(e^{c}\right)}\right) + \log \left(e^{b}\right)\right) + \log \left(e^{a}\right) \]

5. Applied add-log-exp_binary64 0.4

   \[\leadsto \left(\left(\left(e + \color{blue}{\log \left(e^{d}\right)}\right) + \log \left(e^{c}\right)\right) + \log \left(e^{b}\right)\right) + \log \left(e^{a}\right) \]

6. Applied add-log-exp_binary64 0.4

   \[\leadsto \left(\left(\left(\color{blue}{\log \left(e^{e}\right)} + \log \left(e^{d}\right)\right) + \log \left(e^{c}\right)\right) + \log \left(e^{b}\right)\right) + \log \left(e^{a}\right) \]

7. Applied sum-log_binary64 0.4

   \[\leadsto \left(\left(\color{blue}{\log \left(e^{e} \cdot e^{d}\right)} + \log \left(e^{c}\right)\right) + \log \left(e^{b}\right)\right) + \log \left(e^{a}\right) \]

8. Applied sum-log_binary64 0.3

   \[\leadsto \left(\color{blue}{\log \left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right)} + \log \left(e^{b}\right)\right) + \log \left(e^{a}\right) \]

9. Applied sum-log_binary64 0.2

   \[\leadsto \color{blue}{\log \left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot e^{b}\right)} + \log \left(e^{a}\right) \]

10. Applied sum-log_binary64 0.0

    \[\leadsto \color{blue}{\log \left(\left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot e^{b}\right) \cdot e^{a}\right)} \]

11. Final simplification 0.0

    \[\leadsto \log \left(\left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot e^{b}\right) \cdot e^{a}\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 a) (<= a 2.0)) (<= 2.0 b)) (<= b 4.0)) (<= 4.0 c)) (<= c 8.0)) (<= 8.0 d)) (<= d 16.0)) (<= 16.0 e)) (<= e 32.0))

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))